ABCF->ab-angle a

Percentage Accurate: 19.4% → 59.5%

Time: 27.3s

Alternatives: 26

Speedup: 5.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 19.4% 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: 59.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := B\_m \cdot B\_m + A \cdot A\\ t_1 := \mathsf{hypot}\left(B\_m, A - C\right)\\ t_2 := B\_m \cdot B\_m + -4 \cdot \left(A \cdot C\right)\\ t_3 := A + \left(C + t\_1\right)\\ t_4 := \sqrt{\frac{1}{t\_0}}\\ t_5 := 1 - A \cdot t\_4\\ t_6 := \left(4 \cdot A\right) \cdot C\\ t_7 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_6\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_6 - {B\_m}^{2}}\\ t_8 := t\_6 - B\_m \cdot B\_m\\ t_9 := A + \mathsf{hypot}\left(B\_m, A\right)\\ \mathbf{if}\;t\_7 \leq -\infty:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(A + C\right) + t\_1}{t\_2}} \cdot \left(0 - \sqrt{2}\right)\\ \mathbf{elif}\;t\_7 \leq -2 \cdot 10^{-194}:\\ \;\;\;\;\frac{1}{\frac{t\_8}{\sqrt{t\_2 \cdot \left(\left(2 \cdot F\right) \cdot t\_3\right)}}}\\ \mathbf{elif}\;t\_7 \leq 5 \cdot 10^{-113}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(F \cdot \left(B\_m \cdot B\_m\right)\right) \cdot t\_9\right) + C \cdot \left(2 \cdot \left(\left(C \cdot F\right) \cdot \left(\left(A \cdot -4\right) \cdot t\_5 + 0.5 \cdot \left(\left(B\_m \cdot B\_m\right) \cdot \left(t\_4 \cdot \left(1 - \frac{A \cdot A}{t\_0}\right)\right)\right)\right) + F \cdot \left(t\_9 \cdot \left(A \cdot -4\right) + \left(B\_m \cdot B\_m\right) \cdot t\_5\right)\right)\right)}}{t\_8}\\ \mathbf{elif}\;t\_7 \leq \infty:\\ \;\;\;\;\frac{\sqrt{F \cdot t\_2} \cdot \sqrt{2 \cdot t\_3}}{t\_8}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B\_m, C\right)} \cdot \frac{{\left({2}^{0.25}\right)}^{2}}{0 - B\_m}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (+ (* B_m B_m) (* A A)))
        (t_1 (hypot B_m (- A C)))
        (t_2 (+ (* B_m B_m) (* -4.0 (* A C))))
        (t_3 (+ A (+ C t_1)))
        (t_4 (sqrt (/ 1.0 t_0)))
        (t_5 (- 1.0 (* A t_4)))
        (t_6 (* (* 4.0 A) C))
        (t_7
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_6) F))
            (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_6 (pow B_m 2.0))))
        (t_8 (- t_6 (* B_m B_m)))
        (t_9 (+ A (hypot B_m A))))
   (if (<= t_7 (- INFINITY))
     (* (sqrt (* F (/ (+ (+ A C) t_1) t_2))) (- 0.0 (sqrt 2.0)))
     (if (<= t_7 -2e-194)
       (/ 1.0 (/ t_8 (sqrt (* t_2 (* (* 2.0 F) t_3)))))
       (if (<= t_7 5e-113)
         (/
          (sqrt
           (+
            (* 2.0 (* (* F (* B_m B_m)) t_9))
            (*
             C
             (*
              2.0
              (+
               (*
                (* C F)
                (+
                 (* (* A -4.0) t_5)
                 (* 0.5 (* (* B_m B_m) (* t_4 (- 1.0 (/ (* A A) t_0)))))))
               (* F (+ (* t_9 (* A -4.0)) (* (* B_m B_m) t_5))))))))
          t_8)
         (if (<= t_7 INFINITY)
           (/ (* (sqrt (* F t_2)) (sqrt (* 2.0 t_3))) t_8)
           (*
            (sqrt F)
            (*
             (sqrt (+ C (hypot B_m C)))
             (/ (pow (pow 2.0 0.25) 2.0) (- 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 * A);
	double t_1 = hypot(B_m, (A - C));
	double t_2 = (B_m * B_m) + (-4.0 * (A * C));
	double t_3 = A + (C + t_1);
	double t_4 = sqrt((1.0 / t_0));
	double t_5 = 1.0 - (A * t_4);
	double t_6 = (4.0 * A) * C;
	double t_7 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_6) * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_6 - pow(B_m, 2.0));
	double t_8 = t_6 - (B_m * B_m);
	double t_9 = A + hypot(B_m, A);
	double tmp;
	if (t_7 <= -((double) INFINITY)) {
		tmp = sqrt((F * (((A + C) + t_1) / t_2))) * (0.0 - sqrt(2.0));
	} else if (t_7 <= -2e-194) {
		tmp = 1.0 / (t_8 / sqrt((t_2 * ((2.0 * F) * t_3))));
	} else if (t_7 <= 5e-113) {
		tmp = sqrt(((2.0 * ((F * (B_m * B_m)) * t_9)) + (C * (2.0 * (((C * F) * (((A * -4.0) * t_5) + (0.5 * ((B_m * B_m) * (t_4 * (1.0 - ((A * A) / t_0))))))) + (F * ((t_9 * (A * -4.0)) + ((B_m * B_m) * t_5)))))))) / t_8;
	} else if (t_7 <= ((double) INFINITY)) {
		tmp = (sqrt((F * t_2)) * sqrt((2.0 * t_3))) / t_8;
	} else {
		tmp = sqrt(F) * (sqrt((C + hypot(B_m, C))) * (pow(pow(2.0, 0.25), 2.0) / (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 * A);
	double t_1 = Math.hypot(B_m, (A - C));
	double t_2 = (B_m * B_m) + (-4.0 * (A * C));
	double t_3 = A + (C + t_1);
	double t_4 = Math.sqrt((1.0 / t_0));
	double t_5 = 1.0 - (A * t_4);
	double t_6 = (4.0 * A) * C;
	double t_7 = Math.sqrt(((2.0 * ((Math.pow(B_m, 2.0) - t_6) * F)) * ((A + C) + Math.sqrt((Math.pow(B_m, 2.0) + Math.pow((A - C), 2.0)))))) / (t_6 - Math.pow(B_m, 2.0));
	double t_8 = t_6 - (B_m * B_m);
	double t_9 = A + Math.hypot(B_m, A);
	double tmp;
	if (t_7 <= -Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt((F * (((A + C) + t_1) / t_2))) * (0.0 - Math.sqrt(2.0));
	} else if (t_7 <= -2e-194) {
		tmp = 1.0 / (t_8 / Math.sqrt((t_2 * ((2.0 * F) * t_3))));
	} else if (t_7 <= 5e-113) {
		tmp = Math.sqrt(((2.0 * ((F * (B_m * B_m)) * t_9)) + (C * (2.0 * (((C * F) * (((A * -4.0) * t_5) + (0.5 * ((B_m * B_m) * (t_4 * (1.0 - ((A * A) / t_0))))))) + (F * ((t_9 * (A * -4.0)) + ((B_m * B_m) * t_5)))))))) / t_8;
	} else if (t_7 <= Double.POSITIVE_INFINITY) {
		tmp = (Math.sqrt((F * t_2)) * Math.sqrt((2.0 * t_3))) / t_8;
	} else {
		tmp = Math.sqrt(F) * (Math.sqrt((C + Math.hypot(B_m, C))) * (Math.pow(Math.pow(2.0, 0.25), 2.0) / (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 * A)
	t_1 = math.hypot(B_m, (A - C))
	t_2 = (B_m * B_m) + (-4.0 * (A * C))
	t_3 = A + (C + t_1)
	t_4 = math.sqrt((1.0 / t_0))
	t_5 = 1.0 - (A * t_4)
	t_6 = (4.0 * A) * C
	t_7 = math.sqrt(((2.0 * ((math.pow(B_m, 2.0) - t_6) * F)) * ((A + C) + math.sqrt((math.pow(B_m, 2.0) + math.pow((A - C), 2.0)))))) / (t_6 - math.pow(B_m, 2.0))
	t_8 = t_6 - (B_m * B_m)
	t_9 = A + math.hypot(B_m, A)
	tmp = 0
	if t_7 <= -math.inf:
		tmp = math.sqrt((F * (((A + C) + t_1) / t_2))) * (0.0 - math.sqrt(2.0))
	elif t_7 <= -2e-194:
		tmp = 1.0 / (t_8 / math.sqrt((t_2 * ((2.0 * F) * t_3))))
	elif t_7 <= 5e-113:
		tmp = math.sqrt(((2.0 * ((F * (B_m * B_m)) * t_9)) + (C * (2.0 * (((C * F) * (((A * -4.0) * t_5) + (0.5 * ((B_m * B_m) * (t_4 * (1.0 - ((A * A) / t_0))))))) + (F * ((t_9 * (A * -4.0)) + ((B_m * B_m) * t_5)))))))) / t_8
	elif t_7 <= math.inf:
		tmp = (math.sqrt((F * t_2)) * math.sqrt((2.0 * t_3))) / t_8
	else:
		tmp = math.sqrt(F) * (math.sqrt((C + math.hypot(B_m, C))) * (math.pow(math.pow(2.0, 0.25), 2.0) / (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 * A))
	t_1 = hypot(B_m, Float64(A - C))
	t_2 = Float64(Float64(B_m * B_m) + Float64(-4.0 * Float64(A * C)))
	t_3 = Float64(A + Float64(C + t_1))
	t_4 = sqrt(Float64(1.0 / t_0))
	t_5 = Float64(1.0 - Float64(A * t_4))
	t_6 = Float64(Float64(4.0 * A) * C)
	t_7 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_6) * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_6 - (B_m ^ 2.0)))
	t_8 = Float64(t_6 - Float64(B_m * B_m))
	t_9 = Float64(A + hypot(B_m, A))
	tmp = 0.0
	if (t_7 <= Float64(-Inf))
		tmp = Float64(sqrt(Float64(F * Float64(Float64(Float64(A + C) + t_1) / t_2))) * Float64(0.0 - sqrt(2.0)));
	elseif (t_7 <= -2e-194)
		tmp = Float64(1.0 / Float64(t_8 / sqrt(Float64(t_2 * Float64(Float64(2.0 * F) * t_3)))));
	elseif (t_7 <= 5e-113)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64(F * Float64(B_m * B_m)) * t_9)) + Float64(C * Float64(2.0 * Float64(Float64(Float64(C * F) * Float64(Float64(Float64(A * -4.0) * t_5) + Float64(0.5 * Float64(Float64(B_m * B_m) * Float64(t_4 * Float64(1.0 - Float64(Float64(A * A) / t_0))))))) + Float64(F * Float64(Float64(t_9 * Float64(A * -4.0)) + Float64(Float64(B_m * B_m) * t_5)))))))) / t_8);
	elseif (t_7 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(F * t_2)) * sqrt(Float64(2.0 * t_3))) / t_8);
	else
		tmp = Float64(sqrt(F) * Float64(sqrt(Float64(C + hypot(B_m, C))) * Float64(((2.0 ^ 0.25) ^ 2.0) / 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 * A);
	t_1 = hypot(B_m, (A - C));
	t_2 = (B_m * B_m) + (-4.0 * (A * C));
	t_3 = A + (C + t_1);
	t_4 = sqrt((1.0 / t_0));
	t_5 = 1.0 - (A * t_4);
	t_6 = (4.0 * A) * C;
	t_7 = sqrt(((2.0 * (((B_m ^ 2.0) - t_6) * F)) * ((A + C) + sqrt(((B_m ^ 2.0) + ((A - C) ^ 2.0)))))) / (t_6 - (B_m ^ 2.0));
	t_8 = t_6 - (B_m * B_m);
	t_9 = A + hypot(B_m, A);
	tmp = 0.0;
	if (t_7 <= -Inf)
		tmp = sqrt((F * (((A + C) + t_1) / t_2))) * (0.0 - sqrt(2.0));
	elseif (t_7 <= -2e-194)
		tmp = 1.0 / (t_8 / sqrt((t_2 * ((2.0 * F) * t_3))));
	elseif (t_7 <= 5e-113)
		tmp = sqrt(((2.0 * ((F * (B_m * B_m)) * t_9)) + (C * (2.0 * (((C * F) * (((A * -4.0) * t_5) + (0.5 * ((B_m * B_m) * (t_4 * (1.0 - ((A * A) / t_0))))))) + (F * ((t_9 * (A * -4.0)) + ((B_m * B_m) * t_5)))))))) / t_8;
	elseif (t_7 <= Inf)
		tmp = (sqrt((F * t_2)) * sqrt((2.0 * t_3))) / t_8;
	else
		tmp = sqrt(F) * (sqrt((C + hypot(B_m, C))) * (((2.0 ^ 0.25) ^ 2.0) / (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 * A), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$2 = N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(A + N[(C + t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 / t$95$0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(1.0 - N[(A * t$95$4), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$7 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$6), $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$6 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$8 = N[(t$95$6 - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$9 = N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$7, (-Infinity)], N[(N[Sqrt[N[(F * N[(N[(N[(A + C), $MachinePrecision] + t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(0.0 - N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$7, -2e-194], N[(1.0 / N[(t$95$8 / N[Sqrt[N[(t$95$2 * N[(N[(2.0 * F), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$7, 5e-113], N[(N[Sqrt[N[(N[(2.0 * N[(N[(F * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * t$95$9), $MachinePrecision]), $MachinePrecision] + N[(C * N[(2.0 * N[(N[(N[(C * F), $MachinePrecision] * N[(N[(N[(A * -4.0), $MachinePrecision] * t$95$5), $MachinePrecision] + N[(0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] * N[(t$95$4 * N[(1.0 - N[(N[(A * A), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(F * N[(N[(t$95$9 * N[(A * -4.0), $MachinePrecision]), $MachinePrecision] + N[(N[(B$95$m * B$95$m), $MachinePrecision] * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$8), $MachinePrecision], If[LessEqual[t$95$7, Infinity], N[(N[(N[Sqrt[N[(F * t$95$2), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$8), $MachinePrecision], N[(N[Sqrt[F], $MachinePrecision] * N[(N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[Power[2.0, 0.25], $MachinePrecision], 2.0], $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]]]

Derivation

1. Split input into 5 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 3.2%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

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

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

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

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

   5. Simplified 66.8%

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

   if -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))) < -2.00000000000000004e-194

   1. Initial program 98.3%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 98.3%

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

   6. Applied egg-rr 98.3%

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

   if -2.00000000000000004e-194 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < 4.9999999999999997e-113

   1. Initial program 6.3%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 6.3%

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

   5. Taylor expanded in C around 0

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

   7. Simplified 32.0%

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

   if 4.9999999999999997e-113 < (/.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 49.7%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 58.1%

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

      1. pow2 N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. sqrt-prod N/A

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

      6. pow1/2 N/A

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

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

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

   6. Applied egg-rr 82.8%

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

   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. +-commutative N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. hypot-define N/A

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

      13. hypot-lowering-hypot.f64 14.8%

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

   5. Simplified 14.8%

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

      1. *-commutative N/A

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

      2. pow1/2 N/A

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

      3. unpow-prod-down N/A

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

      4. associate-*l* N/A

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

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

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

      6. pow1/2 N/A

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

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

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

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

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

      9. pow1/2 N/A

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

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

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

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

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

      12. +-commutative N/A

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

      13. hypot-define N/A

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

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

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

      15. mul-1-neg N/A

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

      16. neg-sub0 N/A

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

      17. --lowering--.f64 N/A

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

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

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

   7. Applied egg-rr 26.2%

      \[\leadsto \color{blue}{\sqrt{F} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \left(0 - \frac{\sqrt{2}}{B}\right)\right)} \]
   8. Step-by-step derivation

      1. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left({2}^{\frac{1}{2}}\right), B\right)\right)\right)\right) \]

      2. sqr-pow N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left({2}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {2}^{\left(\frac{\frac{1}{2}}{2}\right)}\right), B\right)\right)\right)\right) \]

      3. pow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left({\left({2}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{2}\right), B\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left({2}^{\left(\frac{\frac{1}{2}}{2}\right)}\right), 2\right), B\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{pow.f64}\left(2, \left(\frac{\frac{1}{2}}{2}\right)\right), 2\right), B\right)\right)\right)\right) \]

      6. metadata-eval 26.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{pow.f64}\left(2, \frac{1}{4}\right), 2\right), B\right)\right)\right)\right) \]

   9. Applied egg-rr 26.3%

      \[\leadsto \sqrt{F} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \left(0 - \frac{\color{blue}{{\left({2}^{0.25}\right)}^{2}}}{B}\right)\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 52.4%

   \[\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{\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)}{B \cdot B + -4 \cdot \left(A \cdot C\right)}} \cdot \left(0 - \sqrt{2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -2 \cdot 10^{-194}:\\ \;\;\;\;\frac{1}{\frac{\left(4 \cdot A\right) \cdot C - B \cdot B}{\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq 5 \cdot 10^{-113}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B\right)\right) \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)\right) + C \cdot \left(2 \cdot \left(\left(C \cdot F\right) \cdot \left(\left(A \cdot -4\right) \cdot \left(1 - A \cdot \sqrt{\frac{1}{B \cdot B + A \cdot A}}\right) + 0.5 \cdot \left(\left(B \cdot B\right) \cdot \left(\sqrt{\frac{1}{B \cdot B + A \cdot A}} \cdot \left(1 - \frac{A \cdot A}{B \cdot B + A \cdot A}\right)\right)\right)\right) + F \cdot \left(\left(A + \mathsf{hypot}\left(B, A\right)\right) \cdot \left(A \cdot -4\right) + \left(B \cdot B\right) \cdot \left(1 - A \cdot \sqrt{\frac{1}{B \cdot B + A \cdot A}}\right)\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)} \cdot \sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \frac{{\left({2}^{0.25}\right)}^{2}}{0 - B}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 56.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(B\_m, A - C\right)\\ t_1 := B\_m \cdot B\_m + -4 \cdot \left(A \cdot C\right)\\ t_2 := \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_2\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2 - {B\_m}^{2}}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(A + C\right) + t\_0}{t\_1}} \cdot \left(0 - \sqrt{2}\right)\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{F \cdot t\_1} \cdot \sqrt{2 \cdot \left(A + \left(C + t\_0\right)\right)}}{t\_2 - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B\_m, C\right)} \cdot \frac{{\left({2}^{0.25}\right)}^{2}}{0 - B\_m}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (hypot B_m (- A C)))
        (t_1 (+ (* B_m B_m) (* -4.0 (* A C))))
        (t_2 (* (* 4.0 A) C))
        (t_3
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_2) F))
            (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_2 (pow B_m 2.0)))))
   (if (<= t_3 (- INFINITY))
     (* (sqrt (* F (/ (+ (+ A C) t_0) t_1))) (- 0.0 (sqrt 2.0)))
     (if (<= t_3 INFINITY)
       (/
        (* (sqrt (* F t_1)) (sqrt (* 2.0 (+ A (+ C t_0)))))
        (- t_2 (* B_m B_m)))
       (*
        (sqrt F)
        (*
         (sqrt (+ C (hypot B_m C)))
         (/ (pow (pow 2.0 0.25) 2.0) (- 0.0 B_m))))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = hypot(B_m, (A - C));
	double t_1 = (B_m * B_m) + (-4.0 * (A * C));
	double t_2 = (4.0 * A) * C;
	double t_3 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_2) * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_2 - pow(B_m, 2.0));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = sqrt((F * (((A + C) + t_0) / t_1))) * (0.0 - sqrt(2.0));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = (sqrt((F * t_1)) * sqrt((2.0 * (A + (C + t_0))))) / (t_2 - (B_m * B_m));
	} else {
		tmp = sqrt(F) * (sqrt((C + hypot(B_m, C))) * (pow(pow(2.0, 0.25), 2.0) / (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 = Math.hypot(B_m, (A - C));
	double t_1 = (B_m * B_m) + (-4.0 * (A * C));
	double t_2 = (4.0 * A) * C;
	double t_3 = Math.sqrt(((2.0 * ((Math.pow(B_m, 2.0) - t_2) * F)) * ((A + C) + Math.sqrt((Math.pow(B_m, 2.0) + Math.pow((A - C), 2.0)))))) / (t_2 - Math.pow(B_m, 2.0));
	double tmp;
	if (t_3 <= -Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt((F * (((A + C) + t_0) / t_1))) * (0.0 - Math.sqrt(2.0));
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = (Math.sqrt((F * t_1)) * Math.sqrt((2.0 * (A + (C + t_0))))) / (t_2 - (B_m * B_m));
	} else {
		tmp = Math.sqrt(F) * (Math.sqrt((C + Math.hypot(B_m, C))) * (Math.pow(Math.pow(2.0, 0.25), 2.0) / (0.0 - B_m)));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = math.hypot(B_m, (A - C))
	t_1 = (B_m * B_m) + (-4.0 * (A * C))
	t_2 = (4.0 * A) * C
	t_3 = math.sqrt(((2.0 * ((math.pow(B_m, 2.0) - t_2) * F)) * ((A + C) + math.sqrt((math.pow(B_m, 2.0) + math.pow((A - C), 2.0)))))) / (t_2 - math.pow(B_m, 2.0))
	tmp = 0
	if t_3 <= -math.inf:
		tmp = math.sqrt((F * (((A + C) + t_0) / t_1))) * (0.0 - math.sqrt(2.0))
	elif t_3 <= math.inf:
		tmp = (math.sqrt((F * t_1)) * math.sqrt((2.0 * (A + (C + t_0))))) / (t_2 - (B_m * B_m))
	else:
		tmp = math.sqrt(F) * (math.sqrt((C + math.hypot(B_m, C))) * (math.pow(math.pow(2.0, 0.25), 2.0) / (0.0 - B_m)))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = hypot(B_m, Float64(A - C))
	t_1 = Float64(Float64(B_m * B_m) + Float64(-4.0 * Float64(A * C)))
	t_2 = Float64(Float64(4.0 * A) * C)
	t_3 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_2) * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_2 - (B_m ^ 2.0)))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(sqrt(Float64(F * Float64(Float64(Float64(A + C) + t_0) / t_1))) * Float64(0.0 - sqrt(2.0)));
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(F * t_1)) * sqrt(Float64(2.0 * Float64(A + Float64(C + t_0))))) / Float64(t_2 - Float64(B_m * B_m)));
	else
		tmp = Float64(sqrt(F) * Float64(sqrt(Float64(C + hypot(B_m, C))) * Float64(((2.0 ^ 0.25) ^ 2.0) / 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 = hypot(B_m, (A - C));
	t_1 = (B_m * B_m) + (-4.0 * (A * C));
	t_2 = (4.0 * A) * C;
	t_3 = sqrt(((2.0 * (((B_m ^ 2.0) - t_2) * F)) * ((A + C) + sqrt(((B_m ^ 2.0) + ((A - C) ^ 2.0)))))) / (t_2 - (B_m ^ 2.0));
	tmp = 0.0;
	if (t_3 <= -Inf)
		tmp = sqrt((F * (((A + C) + t_0) / t_1))) * (0.0 - sqrt(2.0));
	elseif (t_3 <= Inf)
		tmp = (sqrt((F * t_1)) * sqrt((2.0 * (A + (C + t_0))))) / (t_2 - (B_m * B_m));
	else
		tmp = sqrt(F) * (sqrt((C + hypot(B_m, C))) * (((2.0 ^ 0.25) ^ 2.0) / (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[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$1 = N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$2), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$2 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[Sqrt[N[(F * N[(N[(N[(A + C), $MachinePrecision] + t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(0.0 - N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[(N[Sqrt[N[(F * t$95$1), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(A + N[(C + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(t$95$2 - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[F], $MachinePrecision] * N[(N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[Power[2.0, 0.25], $MachinePrecision], 2.0], $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 3.2%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

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

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

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

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

   5. Simplified 66.8%

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

   if -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))) < +inf.0

   1. Initial program 54.0%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 55.9%

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

      1. pow2 N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. sqrt-prod N/A

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

      6. pow1/2 N/A

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

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

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

   6. Applied egg-rr 62.7%

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

   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. +-commutative N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. hypot-define N/A

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

      13. hypot-lowering-hypot.f64 14.8%

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

   5. Simplified 14.8%

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

      1. *-commutative N/A

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

      2. pow1/2 N/A

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

      3. unpow-prod-down N/A

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

      4. associate-*l* N/A

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

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

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

      6. pow1/2 N/A

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

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

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

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

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

      9. pow1/2 N/A

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

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

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

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

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

      12. +-commutative N/A

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

      13. hypot-define N/A

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

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

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

      15. mul-1-neg N/A

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

      16. neg-sub0 N/A

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

      17. --lowering--.f64 N/A

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

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

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

   7. Applied egg-rr 26.2%

      \[\leadsto \color{blue}{\sqrt{F} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \left(0 - \frac{\sqrt{2}}{B}\right)\right)} \]
   8. Step-by-step derivation

      1. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left({2}^{\frac{1}{2}}\right), B\right)\right)\right)\right) \]

      2. sqr-pow N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left({2}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {2}^{\left(\frac{\frac{1}{2}}{2}\right)}\right), B\right)\right)\right)\right) \]

      3. pow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left({\left({2}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{2}\right), B\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left({2}^{\left(\frac{\frac{1}{2}}{2}\right)}\right), 2\right), B\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{pow.f64}\left(2, \left(\frac{\frac{1}{2}}{2}\right)\right), 2\right), B\right)\right)\right)\right) \]

      6. metadata-eval 26.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{pow.f64}\left(2, \frac{1}{4}\right), 2\right), B\right)\right)\right)\right) \]

   9. Applied egg-rr 26.3%

      \[\leadsto \sqrt{F} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \left(0 - \frac{\color{blue}{{\left({2}^{0.25}\right)}^{2}}}{B}\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 49.2%

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

Alternative 3: 54.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 5.7 \cdot 10^{+71}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(B\_m \cdot B\_m + -4 \cdot \left(A \cdot C\right)\right)} \cdot \sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\_m\right)} \cdot \frac{\sqrt{2}}{0 - B\_m}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 5.7e+71)
   (/
    (*
     (sqrt (* F (+ (* B_m B_m) (* -4.0 (* A C)))))
     (sqrt (* 2.0 (+ A (+ C (hypot B_m (- A C)))))))
    (- (* (* 4.0 A) C) (* B_m B_m)))
   (* (sqrt F) (* (sqrt (+ C (hypot C B_m))) (/ (sqrt 2.0) (- 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 <= 5.7e+71) {
		tmp = (sqrt((F * ((B_m * B_m) + (-4.0 * (A * C))))) * sqrt((2.0 * (A + (C + hypot(B_m, (A - C))))))) / (((4.0 * A) * C) - (B_m * B_m));
	} else {
		tmp = sqrt(F) * (sqrt((C + hypot(C, B_m))) * (sqrt(2.0) / (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 <= 5.7e+71) {
		tmp = (Math.sqrt((F * ((B_m * B_m) + (-4.0 * (A * C))))) * Math.sqrt((2.0 * (A + (C + Math.hypot(B_m, (A - C))))))) / (((4.0 * A) * C) - (B_m * B_m));
	} else {
		tmp = Math.sqrt(F) * (Math.sqrt((C + Math.hypot(C, B_m))) * (Math.sqrt(2.0) / (0.0 - B_m)));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 5.7e+71:
		tmp = (math.sqrt((F * ((B_m * B_m) + (-4.0 * (A * C))))) * math.sqrt((2.0 * (A + (C + math.hypot(B_m, (A - C))))))) / (((4.0 * A) * C) - (B_m * B_m))
	else:
		tmp = math.sqrt(F) * (math.sqrt((C + math.hypot(C, B_m))) * (math.sqrt(2.0) / (0.0 - B_m)))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 5.7e+71)
		tmp = Float64(Float64(sqrt(Float64(F * Float64(Float64(B_m * B_m) + Float64(-4.0 * Float64(A * C))))) * sqrt(Float64(2.0 * Float64(A + Float64(C + hypot(B_m, Float64(A - C))))))) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m)));
	else
		tmp = Float64(sqrt(F) * Float64(sqrt(Float64(C + hypot(C, B_m))) * Float64(sqrt(2.0) / 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 <= 5.7e+71)
		tmp = (sqrt((F * ((B_m * B_m) + (-4.0 * (A * C))))) * sqrt((2.0 * (A + (C + hypot(B_m, (A - C))))))) / (((4.0 * A) * C) - (B_m * B_m));
	else
		tmp = sqrt(F) * (sqrt((C + hypot(C, B_m))) * (sqrt(2.0) / (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, 5.7e+71], N[(N[(N[Sqrt[N[(F * N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[F], $MachinePrecision] * N[(N[Sqrt[N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 5.7000000000000001e71

   1. Initial program 26.0%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 31.1%

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

      1. pow2 N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. sqrt-prod N/A

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

      6. pow1/2 N/A

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

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

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

   6. Applied egg-rr 38.1%

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

   if 5.7000000000000001e71 < B

   1. Initial program 5.2%

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

   3. Taylor expanded in A around 0

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

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. hypot-define N/A

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

      13. hypot-lowering-hypot.f64 44.6%

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

   5. Simplified 44.6%

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

      1. *-commutative N/A

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

      2. pow1/2 N/A

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

      3. unpow-prod-down N/A

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

      4. associate-*l* N/A

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

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

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

      6. pow1/2 N/A

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

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

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

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

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

      9. pow1/2 N/A

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

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

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

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

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

      12. +-commutative N/A

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

      13. hypot-define N/A

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

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

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

      15. mul-1-neg N/A

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

      16. neg-sub0 N/A

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

      17. --lowering--.f64 N/A

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

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

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

   7. Applied egg-rr 73.2%

      \[\leadsto \color{blue}{\sqrt{F} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \left(0 - \frac{\sqrt{2}}{B}\right)\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

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

      2. sub0-neg N/A

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

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

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

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

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

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

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

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

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

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

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

      8. +-commutative N/A

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

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

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

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

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

      11. hypot-define N/A

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

      12. hypot-lowering-hypot.f64 73.2%

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

   9. Applied egg-rr 73.2%

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

4. Final simplification 44.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 5.7 \cdot 10^{+71}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)} \cdot \sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \frac{\sqrt{2}}{0 - B}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 53.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(B\_m, A - C\right)\\ t_1 := B\_m \cdot B\_m + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B\_m \leq 5 \cdot 10^{+38}:\\ \;\;\;\;\frac{\sqrt{F \cdot t\_1} \cdot \sqrt{2 \cdot \left(A + \left(C + t\_0\right)\right)}}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{elif}\;B\_m \leq 1.45 \cdot 10^{+149}:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(A + C\right) + t\_0}{t\_1}} \cdot \left(0 - \sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(\sqrt{B\_m - C \cdot \left(-1 - 0.5 \cdot \frac{C}{B\_m}\right)} \cdot \frac{\sqrt{2}}{0 - B\_m}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (hypot B_m (- A C))) (t_1 (+ (* B_m B_m) (* -4.0 (* A C)))))
   (if (<= B_m 5e+38)
     (/
      (* (sqrt (* F t_1)) (sqrt (* 2.0 (+ A (+ C t_0)))))
      (- (* (* 4.0 A) C) (* B_m B_m)))
     (if (<= B_m 1.45e+149)
       (* (sqrt (* F (/ (+ (+ A C) t_0) t_1))) (- 0.0 (sqrt 2.0)))
       (*
        (sqrt F)
        (*
         (sqrt (- B_m (* C (- -1.0 (* 0.5 (/ C B_m))))))
         (/ (sqrt 2.0) (- 0.0 B_m))))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = hypot(B_m, (A - C));
	double t_1 = (B_m * B_m) + (-4.0 * (A * C));
	double tmp;
	if (B_m <= 5e+38) {
		tmp = (sqrt((F * t_1)) * sqrt((2.0 * (A + (C + t_0))))) / (((4.0 * A) * C) - (B_m * B_m));
	} else if (B_m <= 1.45e+149) {
		tmp = sqrt((F * (((A + C) + t_0) / t_1))) * (0.0 - sqrt(2.0));
	} else {
		tmp = sqrt(F) * (sqrt((B_m - (C * (-1.0 - (0.5 * (C / B_m)))))) * (sqrt(2.0) / (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 = Math.hypot(B_m, (A - C));
	double t_1 = (B_m * B_m) + (-4.0 * (A * C));
	double tmp;
	if (B_m <= 5e+38) {
		tmp = (Math.sqrt((F * t_1)) * Math.sqrt((2.0 * (A + (C + t_0))))) / (((4.0 * A) * C) - (B_m * B_m));
	} else if (B_m <= 1.45e+149) {
		tmp = Math.sqrt((F * (((A + C) + t_0) / t_1))) * (0.0 - Math.sqrt(2.0));
	} else {
		tmp = Math.sqrt(F) * (Math.sqrt((B_m - (C * (-1.0 - (0.5 * (C / B_m)))))) * (Math.sqrt(2.0) / (0.0 - B_m)));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = math.hypot(B_m, (A - C))
	t_1 = (B_m * B_m) + (-4.0 * (A * C))
	tmp = 0
	if B_m <= 5e+38:
		tmp = (math.sqrt((F * t_1)) * math.sqrt((2.0 * (A + (C + t_0))))) / (((4.0 * A) * C) - (B_m * B_m))
	elif B_m <= 1.45e+149:
		tmp = math.sqrt((F * (((A + C) + t_0) / t_1))) * (0.0 - math.sqrt(2.0))
	else:
		tmp = math.sqrt(F) * (math.sqrt((B_m - (C * (-1.0 - (0.5 * (C / B_m)))))) * (math.sqrt(2.0) / (0.0 - B_m)))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = hypot(B_m, Float64(A - C))
	t_1 = Float64(Float64(B_m * B_m) + Float64(-4.0 * Float64(A * C)))
	tmp = 0.0
	if (B_m <= 5e+38)
		tmp = Float64(Float64(sqrt(Float64(F * t_1)) * sqrt(Float64(2.0 * Float64(A + Float64(C + t_0))))) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m)));
	elseif (B_m <= 1.45e+149)
		tmp = Float64(sqrt(Float64(F * Float64(Float64(Float64(A + C) + t_0) / t_1))) * Float64(0.0 - sqrt(2.0)));
	else
		tmp = Float64(sqrt(F) * Float64(sqrt(Float64(B_m - Float64(C * Float64(-1.0 - Float64(0.5 * Float64(C / B_m)))))) * Float64(sqrt(2.0) / 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 = hypot(B_m, (A - C));
	t_1 = (B_m * B_m) + (-4.0 * (A * C));
	tmp = 0.0;
	if (B_m <= 5e+38)
		tmp = (sqrt((F * t_1)) * sqrt((2.0 * (A + (C + t_0))))) / (((4.0 * A) * C) - (B_m * B_m));
	elseif (B_m <= 1.45e+149)
		tmp = sqrt((F * (((A + C) + t_0) / t_1))) * (0.0 - sqrt(2.0));
	else
		tmp = sqrt(F) * (sqrt((B_m - (C * (-1.0 - (0.5 * (C / B_m)))))) * (sqrt(2.0) / (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[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$1 = N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 5e+38], N[(N[(N[Sqrt[N[(F * t$95$1), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(A + N[(C + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 1.45e+149], N[(N[Sqrt[N[(F * N[(N[(N[(A + C), $MachinePrecision] + t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(0.0 - N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[F], $MachinePrecision] * N[(N[Sqrt[N[(B$95$m - N[(C * N[(-1.0 - N[(0.5 * N[(C / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if B < 4.9999999999999997e38

   1. Initial program 26.1%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 31.3%

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

      1. pow2 N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. sqrt-prod N/A

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

      6. pow1/2 N/A

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

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

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

   6. Applied egg-rr 37.8%

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

   if 4.9999999999999997e38 < B < 1.4500000000000001e149

   1. Initial program 13.0%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

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

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

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

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

   5. Simplified 74.3%

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

   if 1.4500000000000001e149 < B

   1. Initial program 0.0%

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

   3. Taylor expanded in 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. +-commutative N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. hypot-define N/A

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

      13. hypot-lowering-hypot.f64 39.8%

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

   5. Simplified 39.8%

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

      1. *-commutative N/A

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

      2. pow1/2 N/A

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

      3. unpow-prod-down N/A

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

      4. associate-*l* N/A

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

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

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

      6. pow1/2 N/A

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

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

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

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

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

      9. pow1/2 N/A

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

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

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

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

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

      12. +-commutative N/A

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

      13. hypot-define N/A

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

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

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

      15. mul-1-neg N/A

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

      16. neg-sub0 N/A

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

      17. --lowering--.f64 N/A

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

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

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

   7. Applied egg-rr 79.2%

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

   8. Taylor expanded in C around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(B + C \cdot \left(1 + \frac{1}{2} \cdot \frac{C}{B}\right)\right)}\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right)\right)\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(B, \left(C \cdot \left(1 + \frac{1}{2} \cdot \frac{C}{B}\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(B, \mathsf{*.f64}\left(C, \left(1 + \frac{1}{2} \cdot \frac{C}{B}\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(B, \mathsf{*.f64}\left(C, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \frac{C}{B}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(B, \mathsf{*.f64}\left(C, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{C}{B}\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right)\right)\right) \]

      5. /-lowering-/.f64 74.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(B, \mathsf{*.f64}\left(C, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(C, B\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right)\right)\right) \]

   10. Simplified 74.3%

       \[\leadsto \sqrt{F} \cdot \left(\sqrt{\color{blue}{B + C \cdot \left(1 + 0.5 \cdot \frac{C}{B}\right)}} \cdot \left(0 - \frac{\sqrt{2}}{B}\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 5 \cdot 10^{+38}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)} \cdot \sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;B \leq 1.45 \cdot 10^{+149}:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)}{B \cdot B + -4 \cdot \left(A \cdot C\right)}} \cdot \left(0 - \sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(\sqrt{B - C \cdot \left(-1 - 0.5 \cdot \frac{C}{B}\right)} \cdot \frac{\sqrt{2}}{0 - B}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 49.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(B\_m, A - C\right)\\ t_1 := B\_m \cdot B\_m + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B\_m \leq 2.7 \cdot 10^{-55}:\\ \;\;\;\;\frac{\sqrt{t\_1 \cdot \left(\left(2 \cdot F\right) \cdot \left(C + \left(A + t\_0\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{elif}\;B\_m \leq 4.5 \cdot 10^{+148}:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(A + C\right) + t\_0}{t\_1}} \cdot \left(0 - \sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(\sqrt{B\_m - C \cdot \left(-1 - 0.5 \cdot \frac{C}{B\_m}\right)} \cdot \frac{\sqrt{2}}{0 - B\_m}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (hypot B_m (- A C))) (t_1 (+ (* B_m B_m) (* -4.0 (* A C)))))
   (if (<= B_m 2.7e-55)
     (/
      (sqrt (* t_1 (* (* 2.0 F) (+ C (+ A t_0)))))
      (- (* (* 4.0 A) C) (* B_m B_m)))
     (if (<= B_m 4.5e+148)
       (* (sqrt (* F (/ (+ (+ A C) t_0) t_1))) (- 0.0 (sqrt 2.0)))
       (*
        (sqrt F)
        (*
         (sqrt (- B_m (* C (- -1.0 (* 0.5 (/ C B_m))))))
         (/ (sqrt 2.0) (- 0.0 B_m))))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = hypot(B_m, (A - C));
	double t_1 = (B_m * B_m) + (-4.0 * (A * C));
	double tmp;
	if (B_m <= 2.7e-55) {
		tmp = sqrt((t_1 * ((2.0 * F) * (C + (A + t_0))))) / (((4.0 * A) * C) - (B_m * B_m));
	} else if (B_m <= 4.5e+148) {
		tmp = sqrt((F * (((A + C) + t_0) / t_1))) * (0.0 - sqrt(2.0));
	} else {
		tmp = sqrt(F) * (sqrt((B_m - (C * (-1.0 - (0.5 * (C / B_m)))))) * (sqrt(2.0) / (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 = Math.hypot(B_m, (A - C));
	double t_1 = (B_m * B_m) + (-4.0 * (A * C));
	double tmp;
	if (B_m <= 2.7e-55) {
		tmp = Math.sqrt((t_1 * ((2.0 * F) * (C + (A + t_0))))) / (((4.0 * A) * C) - (B_m * B_m));
	} else if (B_m <= 4.5e+148) {
		tmp = Math.sqrt((F * (((A + C) + t_0) / t_1))) * (0.0 - Math.sqrt(2.0));
	} else {
		tmp = Math.sqrt(F) * (Math.sqrt((B_m - (C * (-1.0 - (0.5 * (C / B_m)))))) * (Math.sqrt(2.0) / (0.0 - B_m)));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = math.hypot(B_m, (A - C))
	t_1 = (B_m * B_m) + (-4.0 * (A * C))
	tmp = 0
	if B_m <= 2.7e-55:
		tmp = math.sqrt((t_1 * ((2.0 * F) * (C + (A + t_0))))) / (((4.0 * A) * C) - (B_m * B_m))
	elif B_m <= 4.5e+148:
		tmp = math.sqrt((F * (((A + C) + t_0) / t_1))) * (0.0 - math.sqrt(2.0))
	else:
		tmp = math.sqrt(F) * (math.sqrt((B_m - (C * (-1.0 - (0.5 * (C / B_m)))))) * (math.sqrt(2.0) / (0.0 - B_m)))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = hypot(B_m, Float64(A - C))
	t_1 = Float64(Float64(B_m * B_m) + Float64(-4.0 * Float64(A * C)))
	tmp = 0.0
	if (B_m <= 2.7e-55)
		tmp = Float64(sqrt(Float64(t_1 * Float64(Float64(2.0 * F) * Float64(C + Float64(A + t_0))))) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m)));
	elseif (B_m <= 4.5e+148)
		tmp = Float64(sqrt(Float64(F * Float64(Float64(Float64(A + C) + t_0) / t_1))) * Float64(0.0 - sqrt(2.0)));
	else
		tmp = Float64(sqrt(F) * Float64(sqrt(Float64(B_m - Float64(C * Float64(-1.0 - Float64(0.5 * Float64(C / B_m)))))) * Float64(sqrt(2.0) / 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 = hypot(B_m, (A - C));
	t_1 = (B_m * B_m) + (-4.0 * (A * C));
	tmp = 0.0;
	if (B_m <= 2.7e-55)
		tmp = sqrt((t_1 * ((2.0 * F) * (C + (A + t_0))))) / (((4.0 * A) * C) - (B_m * B_m));
	elseif (B_m <= 4.5e+148)
		tmp = sqrt((F * (((A + C) + t_0) / t_1))) * (0.0 - sqrt(2.0));
	else
		tmp = sqrt(F) * (sqrt((B_m - (C * (-1.0 - (0.5 * (C / B_m)))))) * (sqrt(2.0) / (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[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$1 = N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 2.7e-55], N[(N[Sqrt[N[(t$95$1 * N[(N[(2.0 * F), $MachinePrecision] * N[(C + N[(A + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 4.5e+148], N[(N[Sqrt[N[(F * N[(N[(N[(A + C), $MachinePrecision] + t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(0.0 - N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[F], $MachinePrecision] * N[(N[Sqrt[N[(B$95$m - N[(C * N[(-1.0 - N[(0.5 * N[(C / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if B < 2.70000000000000004e-55

   1. Initial program 22.9%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 28.1%

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. associate-*r* N/A

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

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

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

   6. Applied egg-rr 25.2%

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

      1. metadata-eval N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. pow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

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

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

   8. Applied egg-rr 30.3%

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

   if 2.70000000000000004e-55 < B < 4.49999999999999994e148

   1. Initial program 35.0%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

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

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

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

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

   5. Simplified 69.9%

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

   if 4.49999999999999994e148 < B

   1. Initial program 0.0%

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

   3. Taylor expanded in 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. +-commutative N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. hypot-define N/A

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

      13. hypot-lowering-hypot.f64 39.8%

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

   5. Simplified 39.8%

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

      1. *-commutative N/A

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

      2. pow1/2 N/A

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

      3. unpow-prod-down N/A

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

      4. associate-*l* N/A

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

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

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

      6. pow1/2 N/A

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

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

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

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

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

      9. pow1/2 N/A

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

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

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

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

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

      12. +-commutative N/A

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

      13. hypot-define N/A

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

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

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

      15. mul-1-neg N/A

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

      16. neg-sub0 N/A

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

      17. --lowering--.f64 N/A

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

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

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

   7. Applied egg-rr 79.2%

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

   8. Taylor expanded in C around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(B + C \cdot \left(1 + \frac{1}{2} \cdot \frac{C}{B}\right)\right)}\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right)\right)\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(B, \left(C \cdot \left(1 + \frac{1}{2} \cdot \frac{C}{B}\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(B, \mathsf{*.f64}\left(C, \left(1 + \frac{1}{2} \cdot \frac{C}{B}\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(B, \mathsf{*.f64}\left(C, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \frac{C}{B}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(B, \mathsf{*.f64}\left(C, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{C}{B}\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right)\right)\right) \]

      5. /-lowering-/.f64 74.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(B, \mathsf{*.f64}\left(C, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(C, B\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right)\right)\right) \]

   10. Simplified 74.3%

       \[\leadsto \sqrt{F} \cdot \left(\sqrt{\color{blue}{B + C \cdot \left(1 + 0.5 \cdot \frac{C}{B}\right)}} \cdot \left(0 - \frac{\sqrt{2}}{B}\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.7 \cdot 10^{-55}:\\ \;\;\;\;\frac{\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;B \leq 4.5 \cdot 10^{+148}:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)}{B \cdot B + -4 \cdot \left(A \cdot C\right)}} \cdot \left(0 - \sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(\sqrt{B - C \cdot \left(-1 - 0.5 \cdot \frac{C}{B}\right)} \cdot \frac{\sqrt{2}}{0 - B}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 49.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m\\ \mathbf{if}\;B\_m \leq 9.2 \cdot 10^{+70}:\\ \;\;\;\;\frac{\sqrt{\left(B\_m \cdot B\_m + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B\_m, A - C\right)\right)\right)\right)}}{t\_0}\\ \mathbf{elif}\;B\_m \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{F} \cdot \left(B\_m \cdot \frac{\sqrt{2 \cdot \left(C + \mathsf{hypot}\left(C, B\_m\right)\right)}}{t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(\sqrt{B\_m - C \cdot \left(-1 - 0.5 \cdot \frac{C}{B\_m}\right)} \cdot \frac{\sqrt{2}}{0 - B\_m}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (* (* 4.0 A) C) (* B_m B_m))))
   (if (<= B_m 9.2e+70)
     (/
      (sqrt
       (*
        (+ (* B_m B_m) (* -4.0 (* A C)))
        (* (* 2.0 F) (+ C (+ A (hypot B_m (- A C)))))))
      t_0)
     (if (<= B_m 1.35e+154)
       (* (sqrt F) (* B_m (/ (sqrt (* 2.0 (+ C (hypot C B_m)))) t_0)))
       (*
        (sqrt F)
        (*
         (sqrt (- B_m (* C (- -1.0 (* 0.5 (/ C B_m))))))
         (/ (sqrt 2.0) (- 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 <= 9.2e+70) {
		tmp = sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((2.0 * F) * (C + (A + hypot(B_m, (A - C))))))) / t_0;
	} else if (B_m <= 1.35e+154) {
		tmp = sqrt(F) * (B_m * (sqrt((2.0 * (C + hypot(C, B_m)))) / t_0));
	} else {
		tmp = sqrt(F) * (sqrt((B_m - (C * (-1.0 - (0.5 * (C / B_m)))))) * (sqrt(2.0) / (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 <= 9.2e+70) {
		tmp = Math.sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((2.0 * F) * (C + (A + Math.hypot(B_m, (A - C))))))) / t_0;
	} else if (B_m <= 1.35e+154) {
		tmp = Math.sqrt(F) * (B_m * (Math.sqrt((2.0 * (C + Math.hypot(C, B_m)))) / t_0));
	} else {
		tmp = Math.sqrt(F) * (Math.sqrt((B_m - (C * (-1.0 - (0.5 * (C / B_m)))))) * (Math.sqrt(2.0) / (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 <= 9.2e+70:
		tmp = math.sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((2.0 * F) * (C + (A + math.hypot(B_m, (A - C))))))) / t_0
	elif B_m <= 1.35e+154:
		tmp = math.sqrt(F) * (B_m * (math.sqrt((2.0 * (C + math.hypot(C, B_m)))) / t_0))
	else:
		tmp = math.sqrt(F) * (math.sqrt((B_m - (C * (-1.0 - (0.5 * (C / B_m)))))) * (math.sqrt(2.0) / (0.0 - B_m)))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m))
	tmp = 0.0
	if (B_m <= 9.2e+70)
		tmp = Float64(sqrt(Float64(Float64(Float64(B_m * B_m) + Float64(-4.0 * Float64(A * C))) * Float64(Float64(2.0 * F) * Float64(C + Float64(A + hypot(B_m, Float64(A - C))))))) / t_0);
	elseif (B_m <= 1.35e+154)
		tmp = Float64(sqrt(F) * Float64(B_m * Float64(sqrt(Float64(2.0 * Float64(C + hypot(C, B_m)))) / t_0)));
	else
		tmp = Float64(sqrt(F) * Float64(sqrt(Float64(B_m - Float64(C * Float64(-1.0 - Float64(0.5 * Float64(C / B_m)))))) * Float64(sqrt(2.0) / 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 <= 9.2e+70)
		tmp = sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((2.0 * F) * (C + (A + hypot(B_m, (A - C))))))) / t_0;
	elseif (B_m <= 1.35e+154)
		tmp = sqrt(F) * (B_m * (sqrt((2.0 * (C + hypot(C, B_m)))) / t_0));
	else
		tmp = sqrt(F) * (sqrt((B_m - (C * (-1.0 - (0.5 * (C / B_m)))))) * (sqrt(2.0) / (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[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 9.2e+70], N[(N[Sqrt[N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * F), $MachinePrecision] * N[(C + N[(A + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[B$95$m, 1.35e+154], N[(N[Sqrt[F], $MachinePrecision] * N[(B$95$m * N[(N[Sqrt[N[(2.0 * N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[F], $MachinePrecision] * N[(N[Sqrt[N[(B$95$m - N[(C * N[(-1.0 - N[(0.5 * N[(C / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 9.19999999999999975e70

   1. Initial program 26.0%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 31.1%

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. associate-*r* N/A

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

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

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

   6. Applied egg-rr 28.5%

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

      1. metadata-eval N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. pow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

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

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

   8. Applied egg-rr 33.1%

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

   if 9.19999999999999975e70 < B < 1.35000000000000003e154

   1. Initial program 13.5%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 13.9%

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

   5. Taylor expanded in A around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. hypot-define N/A

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

      11. hypot-lowering-hypot.f64 14.5%

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

   7. Simplified 14.5%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

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

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

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

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

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

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

      12. pow1/2 N/A

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

      13. *-commutative N/A

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

      14. associate-*l* N/A

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

   9. Applied egg-rr 52.2%

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

       1. associate-*l* N/A

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

       2. *-commutative N/A

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

       3. *-commutative N/A

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

       4. unpow1/2 N/A

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

       5. associate-*l* N/A

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

       6. sqrt-prod N/A

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

       7. associate-*l* N/A

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

       8. *-commutative N/A

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

       9. associate-/r/ N/A

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

       10. clear-num N/A

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

   11. Applied egg-rr 63.7%

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

   if 1.35000000000000003e154 < B

   1. Initial program 0.0%

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

   3. Taylor expanded in 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. +-commutative N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. hypot-define N/A

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

      13. hypot-lowering-hypot.f64 39.8%

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

   5. Simplified 39.8%

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

      1. *-commutative N/A

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

      2. pow1/2 N/A

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

      3. unpow-prod-down N/A

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

      4. associate-*l* N/A

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

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

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

      6. pow1/2 N/A

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

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

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

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

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

      9. pow1/2 N/A

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

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

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

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

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

      12. +-commutative N/A

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

      13. hypot-define N/A

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

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

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

      15. mul-1-neg N/A

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

      16. neg-sub0 N/A

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

      17. --lowering--.f64 N/A

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

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

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

   7. Applied egg-rr 79.2%

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

   8. Taylor expanded in C around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(B + C \cdot \left(1 + \frac{1}{2} \cdot \frac{C}{B}\right)\right)}\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right)\right)\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(B, \left(C \cdot \left(1 + \frac{1}{2} \cdot \frac{C}{B}\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(B, \mathsf{*.f64}\left(C, \left(1 + \frac{1}{2} \cdot \frac{C}{B}\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(B, \mathsf{*.f64}\left(C, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \frac{C}{B}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(B, \mathsf{*.f64}\left(C, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{C}{B}\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right)\right)\right) \]

      5. /-lowering-/.f64 74.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(B, \mathsf{*.f64}\left(C, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(C, B\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right)\right)\right) \]

   10. Simplified 74.3%

       \[\leadsto \sqrt{F} \cdot \left(\sqrt{\color{blue}{B + C \cdot \left(1 + 0.5 \cdot \frac{C}{B}\right)}} \cdot \left(0 - \frac{\sqrt{2}}{B}\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 39.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 9.2 \cdot 10^{+70}:\\ \;\;\;\;\frac{\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;B \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{F} \cdot \left(B \cdot \frac{\sqrt{2 \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(\sqrt{B - C \cdot \left(-1 - 0.5 \cdot \frac{C}{B}\right)} \cdot \frac{\sqrt{2}}{0 - B}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 47.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 1.35 \cdot 10^{+71}:\\ \;\;\;\;\frac{\sqrt{\left(B\_m \cdot B\_m + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B\_m, A - C\right)\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(\sqrt{B\_m - C \cdot \left(-1 - 0.5 \cdot \frac{C}{B\_m}\right)} \cdot \frac{\sqrt{2}}{0 - B\_m}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 1.35e+71)
   (/
    (sqrt
     (*
      (+ (* B_m B_m) (* -4.0 (* A C)))
      (* (* 2.0 F) (+ C (+ A (hypot B_m (- A C)))))))
    (- (* (* 4.0 A) C) (* B_m B_m)))
   (*
    (sqrt F)
    (*
     (sqrt (- B_m (* C (- -1.0 (* 0.5 (/ C B_m))))))
     (/ (sqrt 2.0) (- 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 <= 1.35e+71) {
		tmp = sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((2.0 * F) * (C + (A + hypot(B_m, (A - C))))))) / (((4.0 * A) * C) - (B_m * B_m));
	} else {
		tmp = sqrt(F) * (sqrt((B_m - (C * (-1.0 - (0.5 * (C / B_m)))))) * (sqrt(2.0) / (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 <= 1.35e+71) {
		tmp = Math.sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((2.0 * F) * (C + (A + Math.hypot(B_m, (A - C))))))) / (((4.0 * A) * C) - (B_m * B_m));
	} else {
		tmp = Math.sqrt(F) * (Math.sqrt((B_m - (C * (-1.0 - (0.5 * (C / B_m)))))) * (Math.sqrt(2.0) / (0.0 - B_m)));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 1.35e+71:
		tmp = math.sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((2.0 * F) * (C + (A + math.hypot(B_m, (A - C))))))) / (((4.0 * A) * C) - (B_m * B_m))
	else:
		tmp = math.sqrt(F) * (math.sqrt((B_m - (C * (-1.0 - (0.5 * (C / B_m)))))) * (math.sqrt(2.0) / (0.0 - B_m)))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 1.35e+71)
		tmp = Float64(sqrt(Float64(Float64(Float64(B_m * B_m) + Float64(-4.0 * Float64(A * C))) * Float64(Float64(2.0 * F) * Float64(C + Float64(A + hypot(B_m, Float64(A - C))))))) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m)));
	else
		tmp = Float64(sqrt(F) * Float64(sqrt(Float64(B_m - Float64(C * Float64(-1.0 - Float64(0.5 * Float64(C / B_m)))))) * Float64(sqrt(2.0) / 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 <= 1.35e+71)
		tmp = sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((2.0 * F) * (C + (A + hypot(B_m, (A - C))))))) / (((4.0 * A) * C) - (B_m * B_m));
	else
		tmp = sqrt(F) * (sqrt((B_m - (C * (-1.0 - (0.5 * (C / B_m)))))) * (sqrt(2.0) / (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, 1.35e+71], N[(N[Sqrt[N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * F), $MachinePrecision] * N[(C + N[(A + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[F], $MachinePrecision] * N[(N[Sqrt[N[(B$95$m - N[(C * N[(-1.0 - N[(0.5 * N[(C / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1.34999999999999998e71

   1. Initial program 26.0%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 31.1%

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. associate-*r* N/A

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

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

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

   6. Applied egg-rr 28.5%

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

      1. metadata-eval N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. pow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

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

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

   8. Applied egg-rr 33.1%

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

   if 1.34999999999999998e71 < B

   1. Initial program 5.2%

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

   3. Taylor expanded in A around 0

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

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. hypot-define N/A

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

      13. hypot-lowering-hypot.f64 44.6%

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

   5. Simplified 44.6%

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

      1. *-commutative N/A

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

      2. pow1/2 N/A

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

      3. unpow-prod-down N/A

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

      4. associate-*l* N/A

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

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

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

      6. pow1/2 N/A

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

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

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

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

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

      9. pow1/2 N/A

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

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

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

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

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

      12. +-commutative N/A

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

      13. hypot-define N/A

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

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

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

      15. mul-1-neg N/A

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

      16. neg-sub0 N/A

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

      17. --lowering--.f64 N/A

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

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

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

   7. Applied egg-rr 73.2%

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

   8. Taylor expanded in C around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(B + C \cdot \left(1 + \frac{1}{2} \cdot \frac{C}{B}\right)\right)}\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right)\right)\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(B, \left(C \cdot \left(1 + \frac{1}{2} \cdot \frac{C}{B}\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(B, \mathsf{*.f64}\left(C, \left(1 + \frac{1}{2} \cdot \frac{C}{B}\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(B, \mathsf{*.f64}\left(C, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \frac{C}{B}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(B, \mathsf{*.f64}\left(C, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{C}{B}\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right)\right)\right) \]

      5. /-lowering-/.f64 64.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(B, \mathsf{*.f64}\left(C, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(C, B\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right)\right)\right) \]

   10. Simplified 64.1%

       \[\leadsto \sqrt{F} \cdot \left(\sqrt{\color{blue}{B + C \cdot \left(1 + 0.5 \cdot \frac{C}{B}\right)}} \cdot \left(0 - \frac{\sqrt{2}}{B}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.35 \cdot 10^{+71}:\\ \;\;\;\;\frac{\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(\sqrt{B - C \cdot \left(-1 - 0.5 \cdot \frac{C}{B}\right)} \cdot \frac{\sqrt{2}}{0 - B}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 48.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 4.5 \cdot 10^{+71}:\\ \;\;\;\;\frac{\sqrt{\left(B\_m \cdot B\_m + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B\_m, A - C\right)\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(\frac{\sqrt{2}}{0 - B\_m} \cdot \sqrt{B\_m + C}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 4.5e+71)
   (/
    (sqrt
     (*
      (+ (* B_m B_m) (* -4.0 (* A C)))
      (* (* 2.0 F) (+ C (+ A (hypot B_m (- A C)))))))
    (- (* (* 4.0 A) C) (* B_m B_m)))
   (* (sqrt F) (* (/ (sqrt 2.0) (- 0.0 B_m)) (sqrt (+ B_m C))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 4.5e+71) {
		tmp = sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((2.0 * F) * (C + (A + hypot(B_m, (A - C))))))) / (((4.0 * A) * C) - (B_m * B_m));
	} else {
		tmp = sqrt(F) * ((sqrt(2.0) / (0.0 - B_m)) * sqrt((B_m + C)));
	}
	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 <= 4.5e+71) {
		tmp = Math.sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((2.0 * F) * (C + (A + Math.hypot(B_m, (A - C))))))) / (((4.0 * A) * C) - (B_m * B_m));
	} else {
		tmp = Math.sqrt(F) * ((Math.sqrt(2.0) / (0.0 - B_m)) * Math.sqrt((B_m + C)));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 4.5e+71:
		tmp = math.sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((2.0 * F) * (C + (A + math.hypot(B_m, (A - C))))))) / (((4.0 * A) * C) - (B_m * B_m))
	else:
		tmp = math.sqrt(F) * ((math.sqrt(2.0) / (0.0 - B_m)) * math.sqrt((B_m + C)))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 4.5e+71)
		tmp = Float64(sqrt(Float64(Float64(Float64(B_m * B_m) + Float64(-4.0 * Float64(A * C))) * Float64(Float64(2.0 * F) * Float64(C + Float64(A + hypot(B_m, Float64(A - C))))))) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m)));
	else
		tmp = Float64(sqrt(F) * Float64(Float64(sqrt(2.0) / Float64(0.0 - B_m)) * sqrt(Float64(B_m + C))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 4.5e+71)
		tmp = sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((2.0 * F) * (C + (A + hypot(B_m, (A - C))))))) / (((4.0 * A) * C) - (B_m * B_m));
	else
		tmp = sqrt(F) * ((sqrt(2.0) / (0.0 - B_m)) * sqrt((B_m + C)));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 4.5e+71], N[(N[Sqrt[N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * F), $MachinePrecision] * N[(C + N[(A + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[F], $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 4.50000000000000043e71

   1. Initial program 26.0%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 31.1%

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. associate-*r* N/A

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

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

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

   6. Applied egg-rr 28.5%

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

      1. metadata-eval N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. pow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

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

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

   8. Applied egg-rr 33.1%

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

   if 4.50000000000000043e71 < B

   1. Initial program 5.2%

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

   3. Taylor expanded in A around 0

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

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. hypot-define N/A

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

      13. hypot-lowering-hypot.f64 44.6%

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

   5. Simplified 44.6%

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

      1. *-commutative N/A

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

      2. pow1/2 N/A

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

      3. unpow-prod-down N/A

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

      4. associate-*l* N/A

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

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

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

      6. pow1/2 N/A

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

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

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

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

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

      9. pow1/2 N/A

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

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

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

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

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

      12. +-commutative N/A

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

      13. hypot-define N/A

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

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

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

      15. mul-1-neg N/A

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

      16. neg-sub0 N/A

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

      17. --lowering--.f64 N/A

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

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

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

   7. Applied egg-rr 73.2%

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

   8. Taylor expanded in C around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(B + C\right)}\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 63.8%

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

   10. Simplified 63.8%

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

4. Final simplification 38.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 4.5 \cdot 10^{+71}:\\ \;\;\;\;\frac{\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(\frac{\sqrt{2}}{0 - B} \cdot \sqrt{B + C}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 47.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 1.32 \cdot 10^{+71}:\\ \;\;\;\;\frac{\sqrt{\left(B\_m \cdot B\_m + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B\_m, A - C\right)\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(\sqrt{\frac{1}{B\_m}} \cdot \left(0 - \sqrt{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 1.32e+71)
   (/
    (sqrt
     (*
      (+ (* B_m B_m) (* -4.0 (* A C)))
      (* (* 2.0 F) (+ C (+ A (hypot B_m (- A C)))))))
    (- (* (* 4.0 A) C) (* B_m B_m)))
   (* (sqrt F) (* (sqrt (/ 1.0 B_m)) (- 0.0 (sqrt 2.0))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.32e+71) {
		tmp = sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((2.0 * F) * (C + (A + hypot(B_m, (A - C))))))) / (((4.0 * A) * C) - (B_m * B_m));
	} else {
		tmp = sqrt(F) * (sqrt((1.0 / B_m)) * (0.0 - sqrt(2.0)));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.32e+71) {
		tmp = Math.sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((2.0 * F) * (C + (A + Math.hypot(B_m, (A - C))))))) / (((4.0 * A) * C) - (B_m * B_m));
	} else {
		tmp = Math.sqrt(F) * (Math.sqrt((1.0 / B_m)) * (0.0 - Math.sqrt(2.0)));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 1.32e+71:
		tmp = math.sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((2.0 * F) * (C + (A + math.hypot(B_m, (A - C))))))) / (((4.0 * A) * C) - (B_m * B_m))
	else:
		tmp = math.sqrt(F) * (math.sqrt((1.0 / B_m)) * (0.0 - math.sqrt(2.0)))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 1.32e+71)
		tmp = Float64(sqrt(Float64(Float64(Float64(B_m * B_m) + Float64(-4.0 * Float64(A * C))) * Float64(Float64(2.0 * F) * Float64(C + Float64(A + hypot(B_m, Float64(A - C))))))) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m)));
	else
		tmp = Float64(sqrt(F) * Float64(sqrt(Float64(1.0 / B_m)) * Float64(0.0 - sqrt(2.0))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 1.32e+71)
		tmp = sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((2.0 * F) * (C + (A + hypot(B_m, (A - C))))))) / (((4.0 * A) * C) - (B_m * B_m));
	else
		tmp = sqrt(F) * (sqrt((1.0 / B_m)) * (0.0 - sqrt(2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 1.32e+71], N[(N[Sqrt[N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * F), $MachinePrecision] * N[(C + N[(A + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[F], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / B$95$m), $MachinePrecision]], $MachinePrecision] * N[(0.0 - N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1.32000000000000007e71

   1. Initial program 26.0%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 31.1%

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. associate-*r* N/A

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

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

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

   6. Applied egg-rr 28.5%

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

      1. metadata-eval N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. pow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

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

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

   8. Applied egg-rr 33.1%

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

   if 1.32000000000000007e71 < B

   1. Initial program 5.2%

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

   3. Taylor expanded in A around 0

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

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. hypot-define N/A

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

      13. hypot-lowering-hypot.f64 44.6%

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

   5. Simplified 44.6%

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

      1. *-commutative N/A

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

      2. pow1/2 N/A

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

      3. unpow-prod-down N/A

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

      4. associate-*l* N/A

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

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

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

      6. pow1/2 N/A

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

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

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

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

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

      9. pow1/2 N/A

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

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

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

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

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

      12. +-commutative N/A

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

      13. hypot-define N/A

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

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

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

      15. mul-1-neg N/A

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

      16. neg-sub0 N/A

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

      17. --lowering--.f64 N/A

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

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

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

   7. Applied egg-rr 73.2%

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

   8. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

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

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

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

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

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

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

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

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

      6. sqrt-lowering-sqrt.f64 61.0%

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

   10. Simplified 61.0%

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

4. Final simplification 38.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.32 \cdot 10^{+71}:\\ \;\;\;\;\frac{\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(\sqrt{\frac{1}{B}} \cdot \left(0 - \sqrt{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 39.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 6.5 \cdot 10^{+74}:\\ \;\;\;\;\frac{\sqrt{\left(B\_m \cdot B\_m + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B\_m, A - C\right)\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{0 - B\_m} \cdot \sqrt{F \cdot \left(B\_m + C\right)}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 6.5e+74)
   (/
    (sqrt
     (*
      (+ (* B_m B_m) (* -4.0 (* A C)))
      (* (* 2.0 F) (+ C (+ A (hypot B_m (- A C)))))))
    (- (* (* 4.0 A) C) (* B_m B_m)))
   (* (/ (sqrt 2.0) (- 0.0 B_m)) (sqrt (* F (+ B_m C))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 6.5e+74) {
		tmp = sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((2.0 * F) * (C + (A + hypot(B_m, (A - C))))))) / (((4.0 * A) * C) - (B_m * B_m));
	} else {
		tmp = (sqrt(2.0) / (0.0 - B_m)) * sqrt((F * (B_m + C)));
	}
	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 <= 6.5e+74) {
		tmp = Math.sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((2.0 * F) * (C + (A + Math.hypot(B_m, (A - C))))))) / (((4.0 * A) * C) - (B_m * B_m));
	} else {
		tmp = (Math.sqrt(2.0) / (0.0 - B_m)) * Math.sqrt((F * (B_m + C)));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 6.5e+74:
		tmp = math.sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((2.0 * F) * (C + (A + math.hypot(B_m, (A - C))))))) / (((4.0 * A) * C) - (B_m * B_m))
	else:
		tmp = (math.sqrt(2.0) / (0.0 - B_m)) * math.sqrt((F * (B_m + C)))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 6.5e+74)
		tmp = Float64(sqrt(Float64(Float64(Float64(B_m * B_m) + Float64(-4.0 * Float64(A * C))) * Float64(Float64(2.0 * F) * Float64(C + Float64(A + hypot(B_m, Float64(A - C))))))) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m)));
	else
		tmp = Float64(Float64(sqrt(2.0) / Float64(0.0 - B_m)) * sqrt(Float64(F * Float64(B_m + C))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 6.5e+74)
		tmp = sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((2.0 * F) * (C + (A + hypot(B_m, (A - C))))))) / (((4.0 * A) * C) - (B_m * B_m));
	else
		tmp = (sqrt(2.0) / (0.0 - B_m)) * sqrt((F * (B_m + C)));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 6.5e+74], N[(N[Sqrt[N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * F), $MachinePrecision] * N[(C + N[(A + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(F * N[(B$95$m + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 6.49999999999999962e74

   1. Initial program 25.9%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 31.0%

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. associate-*r* N/A

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

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

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

   6. Applied egg-rr 28.4%

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

      1. metadata-eval N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. pow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

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

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

   8. Applied egg-rr 33.0%

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

   if 6.49999999999999962e74 < B

   1. Initial program 5.3%

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

   3. Taylor expanded in B around inf

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

   5. Simplified 3.4%

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

   6. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      8. +-lowering-+.f64 37.2%

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

   8. Simplified 37.2%

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

4. Final simplification 33.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 6.5 \cdot 10^{+74}:\\ \;\;\;\;\frac{\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{0 - B} \cdot \sqrt{F \cdot \left(B + C\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 38.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;B\_m \leq 2 \cdot 10^{+67}:\\ \;\;\;\;\frac{\sqrt{\left(\left(2 \cdot F\right) \cdot \left(B\_m \cdot B\_m - t\_0\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B\_m, A - C\right)\right)}}{t\_0 - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{0 - B\_m} \cdot \sqrt{F \cdot \left(B\_m + C\right)}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C)))
   (if (<= B_m 2e+67)
     (/
      (sqrt
       (* (* (* 2.0 F) (- (* B_m B_m) t_0)) (+ (+ A C) (hypot B_m (- A C)))))
      (- t_0 (* B_m B_m)))
     (* (/ (sqrt 2.0) (- 0.0 B_m)) (sqrt (* F (+ B_m C)))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double tmp;
	if (B_m <= 2e+67) {
		tmp = sqrt((((2.0 * F) * ((B_m * B_m) - t_0)) * ((A + C) + hypot(B_m, (A - C))))) / (t_0 - (B_m * B_m));
	} else {
		tmp = (sqrt(2.0) / (0.0 - B_m)) * sqrt((F * (B_m + C)));
	}
	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;
	double tmp;
	if (B_m <= 2e+67) {
		tmp = Math.sqrt((((2.0 * F) * ((B_m * B_m) - t_0)) * ((A + C) + Math.hypot(B_m, (A - C))))) / (t_0 - (B_m * B_m));
	} else {
		tmp = (Math.sqrt(2.0) / (0.0 - B_m)) * Math.sqrt((F * (B_m + C)));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = (4.0 * A) * C
	tmp = 0
	if B_m <= 2e+67:
		tmp = math.sqrt((((2.0 * F) * ((B_m * B_m) - t_0)) * ((A + C) + math.hypot(B_m, (A - C))))) / (t_0 - (B_m * B_m))
	else:
		tmp = (math.sqrt(2.0) / (0.0 - B_m)) * math.sqrt((F * (B_m + C)))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	tmp = 0.0
	if (B_m <= 2e+67)
		tmp = Float64(sqrt(Float64(Float64(Float64(2.0 * F) * Float64(Float64(B_m * B_m) - t_0)) * Float64(Float64(A + C) + hypot(B_m, Float64(A - C))))) / Float64(t_0 - Float64(B_m * B_m)));
	else
		tmp = Float64(Float64(sqrt(2.0) / Float64(0.0 - B_m)) * sqrt(Float64(F * Float64(B_m + C))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = (4.0 * A) * C;
	tmp = 0.0;
	if (B_m <= 2e+67)
		tmp = sqrt((((2.0 * F) * ((B_m * B_m) - t_0)) * ((A + C) + hypot(B_m, (A - C))))) / (t_0 - (B_m * B_m));
	else
		tmp = (sqrt(2.0) / (0.0 - B_m)) * sqrt((F * (B_m + C)));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, If[LessEqual[B$95$m, 2e+67], N[(N[Sqrt[N[(N[(N[(2.0 * F), $MachinePrecision] * N[(N[(B$95$m * B$95$m), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(F * N[(B$95$m + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 1.99999999999999997e67

   1. Initial program 26.0%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 31.1%

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

   if 1.99999999999999997e67 < B

   1. Initial program 5.2%

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

   3. Taylor expanded in B around inf

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

   5. Simplified 3.3%

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

   6. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      8. +-lowering-+.f64 36.5%

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

   8. Simplified 36.5%

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

4. Final simplification 32.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2 \cdot 10^{+67}:\\ \;\;\;\;\frac{\sqrt{\left(\left(2 \cdot F\right) \cdot \left(B \cdot B - \left(4 \cdot A\right) \cdot C\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{0 - B} \cdot \sqrt{F \cdot \left(B + C\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 40.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;F \leq 9.2 \cdot 10^{-308}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A + C\right) + \mathsf{hypot}\left(B\_m, A - C\right)\right) \cdot \left(\left(-4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{elif}\;F \leq 7.5 \cdot 10^{+55}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(0 - \sqrt{B\_m \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B\_m}} \cdot \left(0 - \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F 9.2e-308)
   (/
    (sqrt (* (+ (+ A C) (hypot B_m (- A C))) (* (* -4.0 (* A C)) (* 2.0 F))))
    (- (* (* 4.0 A) C) (* B_m B_m)))
   (if (<= F 7.5e+55)
     (* (/ (sqrt 2.0) B_m) (- 0.0 (sqrt (* B_m F))))
     (* (sqrt (/ F B_m)) (- 0.0 (sqrt 2.0))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 9.2e-308) {
		tmp = sqrt((((A + C) + hypot(B_m, (A - C))) * ((-4.0 * (A * C)) * (2.0 * F)))) / (((4.0 * A) * C) - (B_m * B_m));
	} else if (F <= 7.5e+55) {
		tmp = (sqrt(2.0) / B_m) * (0.0 - sqrt((B_m * F)));
	} else {
		tmp = sqrt((F / B_m)) * (0.0 - sqrt(2.0));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 9.2e-308) {
		tmp = Math.sqrt((((A + C) + Math.hypot(B_m, (A - C))) * ((-4.0 * (A * C)) * (2.0 * F)))) / (((4.0 * A) * C) - (B_m * B_m));
	} else if (F <= 7.5e+55) {
		tmp = (Math.sqrt(2.0) / B_m) * (0.0 - Math.sqrt((B_m * F)));
	} else {
		tmp = Math.sqrt((F / B_m)) * (0.0 - Math.sqrt(2.0));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if F <= 9.2e-308:
		tmp = math.sqrt((((A + C) + math.hypot(B_m, (A - C))) * ((-4.0 * (A * C)) * (2.0 * F)))) / (((4.0 * A) * C) - (B_m * B_m))
	elif F <= 7.5e+55:
		tmp = (math.sqrt(2.0) / B_m) * (0.0 - math.sqrt((B_m * F)))
	else:
		tmp = math.sqrt((F / B_m)) * (0.0 - math.sqrt(2.0))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= 9.2e-308)
		tmp = Float64(sqrt(Float64(Float64(Float64(A + C) + hypot(B_m, Float64(A - C))) * Float64(Float64(-4.0 * Float64(A * C)) * Float64(2.0 * F)))) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m)));
	elseif (F <= 7.5e+55)
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(0.0 - sqrt(Float64(B_m * F))));
	else
		tmp = Float64(sqrt(Float64(F / B_m)) * Float64(0.0 - sqrt(2.0)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= 9.2e-308)
		tmp = sqrt((((A + C) + hypot(B_m, (A - C))) * ((-4.0 * (A * C)) * (2.0 * F)))) / (((4.0 * A) * C) - (B_m * B_m));
	elseif (F <= 7.5e+55)
		tmp = (sqrt(2.0) / B_m) * (0.0 - sqrt((B_m * F)));
	else
		tmp = sqrt((F / B_m)) * (0.0 - sqrt(2.0));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, 9.2e-308], N[(N[Sqrt[N[(N[(N[(A + C), $MachinePrecision] + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7.5e+55], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(0.0 - N[Sqrt[N[(B$95$m * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision] * N[(0.0 - N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < 9.1999999999999996e-308

   1. Initial program 34.2%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 39.4%

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

   5. Taylor expanded in B around 0

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

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

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

      2. *-lowering-*.f64 39.4%

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

   7. Simplified 39.4%

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

   if 9.1999999999999996e-308 < F < 7.50000000000000014e55

   1. Initial program 22.1%

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

   3. Taylor expanded in A around 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. +-commutative N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. hypot-define N/A

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

      13. hypot-lowering-hypot.f64 21.7%

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

   5. Simplified 21.7%

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

   6. Taylor expanded in C around 0

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

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

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

      2. *-lowering-*.f64 16.4%

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

   8. Simplified 16.4%

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

   if 7.50000000000000014e55 < F

   1. Initial program 17.4%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

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

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

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

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

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

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

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

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

      6. sqrt-lowering-sqrt.f64 22.3%

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

   5. Simplified 22.3%

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

4. Final simplification 21.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 9.2 \cdot 10^{-308}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(\left(-4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;F \leq 7.5 \cdot 10^{+55}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(0 - \sqrt{B \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(0 - \sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 39.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;F \leq -2.15 \cdot 10^{-246}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A + C\right) + \mathsf{hypot}\left(B\_m, A - C\right)\right) \cdot \left(\left(C \cdot F\right) \cdot \left(A \cdot -8\right)\right)}}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{elif}\;F \leq 7.5 \cdot 10^{+55}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(0 - \sqrt{B\_m \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B\_m}} \cdot \left(0 - \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F -2.15e-246)
   (/
    (sqrt (* (+ (+ A C) (hypot B_m (- A C))) (* (* C F) (* A -8.0))))
    (- (* (* 4.0 A) C) (* B_m B_m)))
   (if (<= F 7.5e+55)
     (* (/ (sqrt 2.0) B_m) (- 0.0 (sqrt (* B_m F))))
     (* (sqrt (/ F B_m)) (- 0.0 (sqrt 2.0))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= -2.15e-246) {
		tmp = sqrt((((A + C) + hypot(B_m, (A - C))) * ((C * F) * (A * -8.0)))) / (((4.0 * A) * C) - (B_m * B_m));
	} else if (F <= 7.5e+55) {
		tmp = (sqrt(2.0) / B_m) * (0.0 - sqrt((B_m * F)));
	} else {
		tmp = sqrt((F / B_m)) * (0.0 - sqrt(2.0));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= -2.15e-246) {
		tmp = Math.sqrt((((A + C) + Math.hypot(B_m, (A - C))) * ((C * F) * (A * -8.0)))) / (((4.0 * A) * C) - (B_m * B_m));
	} else if (F <= 7.5e+55) {
		tmp = (Math.sqrt(2.0) / B_m) * (0.0 - Math.sqrt((B_m * F)));
	} else {
		tmp = Math.sqrt((F / B_m)) * (0.0 - Math.sqrt(2.0));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if F <= -2.15e-246:
		tmp = math.sqrt((((A + C) + math.hypot(B_m, (A - C))) * ((C * F) * (A * -8.0)))) / (((4.0 * A) * C) - (B_m * B_m))
	elif F <= 7.5e+55:
		tmp = (math.sqrt(2.0) / B_m) * (0.0 - math.sqrt((B_m * F)))
	else:
		tmp = math.sqrt((F / B_m)) * (0.0 - math.sqrt(2.0))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= -2.15e-246)
		tmp = Float64(sqrt(Float64(Float64(Float64(A + C) + hypot(B_m, Float64(A - C))) * Float64(Float64(C * F) * Float64(A * -8.0)))) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m)));
	elseif (F <= 7.5e+55)
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(0.0 - sqrt(Float64(B_m * F))));
	else
		tmp = Float64(sqrt(Float64(F / B_m)) * Float64(0.0 - sqrt(2.0)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= -2.15e-246)
		tmp = sqrt((((A + C) + hypot(B_m, (A - C))) * ((C * F) * (A * -8.0)))) / (((4.0 * A) * C) - (B_m * B_m));
	elseif (F <= 7.5e+55)
		tmp = (sqrt(2.0) / B_m) * (0.0 - sqrt((B_m * F)));
	else
		tmp = sqrt((F / B_m)) * (0.0 - sqrt(2.0));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, -2.15e-246], N[(N[Sqrt[N[(N[(N[(A + C), $MachinePrecision] + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(C * F), $MachinePrecision] * N[(A * -8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7.5e+55], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(0.0 - N[Sqrt[N[(B$95$m * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision] * N[(0.0 - N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.14999999999999996e-246

   1. Initial program 40.2%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 43.3%

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

   5. Taylor expanded in B around 0

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

      1. associate-*r* N/A

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

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

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

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

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

      4. *-lowering-*.f64 43.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-8, A\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(A, C\right), \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 43.2%

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

   if -2.14999999999999996e-246 < F < 7.50000000000000014e55

   1. Initial program 21.2%

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

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. hypot-define N/A

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

      13. hypot-lowering-hypot.f64 20.8%

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

   5. Simplified 20.8%

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

   6. Taylor expanded in C around 0

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

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

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

      2. *-lowering-*.f64 15.8%

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

   8. Simplified 15.8%

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

   if 7.50000000000000014e55 < F

   1. Initial program 17.4%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

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

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

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

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

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

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

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

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

      6. sqrt-lowering-sqrt.f64 22.3%

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

   5. Simplified 22.3%

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

4. Final simplification 21.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.15 \cdot 10^{-246}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(\left(C \cdot F\right) \cdot \left(A \cdot -8\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;F \leq 7.5 \cdot 10^{+55}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(0 - \sqrt{B \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(0 - \sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 38.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;F \leq 9.5 \cdot 10^{-304}:\\ \;\;\;\;\frac{\sqrt{C \cdot \left(-16 \cdot \left(A \cdot \left(C \cdot F\right)\right) + \left(F \cdot \left(B\_m \cdot B\_m\right)\right) \cdot 8\right)}}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{elif}\;F \leq 7.5 \cdot 10^{+55}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(0 - \sqrt{B\_m \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B\_m}} \cdot \left(0 - \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F 9.5e-304)
   (/
    (sqrt (* C (+ (* -16.0 (* A (* C F))) (* (* F (* B_m B_m)) 8.0))))
    (- (* (* 4.0 A) C) (* B_m B_m)))
   (if (<= F 7.5e+55)
     (* (/ (sqrt 2.0) B_m) (- 0.0 (sqrt (* B_m F))))
     (* (sqrt (/ F B_m)) (- 0.0 (sqrt 2.0))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 9.5e-304) {
		tmp = sqrt((C * ((-16.0 * (A * (C * F))) + ((F * (B_m * B_m)) * 8.0)))) / (((4.0 * A) * C) - (B_m * B_m));
	} else if (F <= 7.5e+55) {
		tmp = (sqrt(2.0) / B_m) * (0.0 - sqrt((B_m * F)));
	} else {
		tmp = sqrt((F / B_m)) * (0.0 - sqrt(2.0));
	}
	return tmp;
}

Fortran

B_m = abs(b)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (f <= 9.5d-304) then
        tmp = sqrt((c * (((-16.0d0) * (a * (c * f))) + ((f * (b_m * b_m)) * 8.0d0)))) / (((4.0d0 * a) * c) - (b_m * b_m))
    else if (f <= 7.5d+55) then
        tmp = (sqrt(2.0d0) / b_m) * (0.0d0 - sqrt((b_m * f)))
    else
        tmp = sqrt((f / b_m)) * (0.0d0 - sqrt(2.0d0))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 9.5e-304) {
		tmp = Math.sqrt((C * ((-16.0 * (A * (C * F))) + ((F * (B_m * B_m)) * 8.0)))) / (((4.0 * A) * C) - (B_m * B_m));
	} else if (F <= 7.5e+55) {
		tmp = (Math.sqrt(2.0) / B_m) * (0.0 - Math.sqrt((B_m * F)));
	} else {
		tmp = Math.sqrt((F / B_m)) * (0.0 - Math.sqrt(2.0));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if F <= 9.5e-304:
		tmp = math.sqrt((C * ((-16.0 * (A * (C * F))) + ((F * (B_m * B_m)) * 8.0)))) / (((4.0 * A) * C) - (B_m * B_m))
	elif F <= 7.5e+55:
		tmp = (math.sqrt(2.0) / B_m) * (0.0 - math.sqrt((B_m * F)))
	else:
		tmp = math.sqrt((F / B_m)) * (0.0 - math.sqrt(2.0))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= 9.5e-304)
		tmp = Float64(sqrt(Float64(C * Float64(Float64(-16.0 * Float64(A * Float64(C * F))) + Float64(Float64(F * Float64(B_m * B_m)) * 8.0)))) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m)));
	elseif (F <= 7.5e+55)
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(0.0 - sqrt(Float64(B_m * F))));
	else
		tmp = Float64(sqrt(Float64(F / B_m)) * Float64(0.0 - sqrt(2.0)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= 9.5e-304)
		tmp = sqrt((C * ((-16.0 * (A * (C * F))) + ((F * (B_m * B_m)) * 8.0)))) / (((4.0 * A) * C) - (B_m * B_m));
	elseif (F <= 7.5e+55)
		tmp = (sqrt(2.0) / B_m) * (0.0 - sqrt((B_m * F)));
	else
		tmp = sqrt((F / B_m)) * (0.0 - sqrt(2.0));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, 9.5e-304], N[(N[Sqrt[N[(C * N[(N[(-16.0 * N[(A * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(F * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7.5e+55], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(0.0 - N[Sqrt[N[(B$95$m * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision] * N[(0.0 - N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < 9.50000000000000023e-304

   1. Initial program 33.3%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 41.0%

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. associate-*r* N/A

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

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

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

   6. Applied egg-rr 38.4%

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

   7. Taylor expanded in A around -inf

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

      1. mul-1-neg N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      8. unpow2 N/A

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

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

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 23.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(A, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), C\right), A\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(C, C\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Simplified 23.6%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-A \cdot \left(-4 \cdot \frac{\left(B \cdot B\right) \cdot C}{A} + 8 \cdot \left(C \cdot C\right)\right)\right)} \cdot \left(2 \cdot F\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   10. Taylor expanded in C around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(C \cdot \left(-16 \cdot \left(A \cdot \left(C \cdot F\right)\right) + 8 \cdot \left({B}^{2} \cdot F\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   11. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(C, \left(-16 \cdot \left(A \cdot \left(C \cdot F\right)\right) + 8 \cdot \left({B}^{2} \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(C, \mathsf{+.f64}\left(\left(-16 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right), \left(8 \cdot \left({B}^{2} \cdot F\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(C, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \left(A \cdot \left(C \cdot F\right)\right)\right), \left(8 \cdot \left({B}^{2} \cdot F\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(C, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \left(\left(C \cdot F\right) \cdot A\right)\right), \left(8 \cdot \left({B}^{2} \cdot F\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(C, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left(C \cdot F\right), A\right)\right), \left(8 \cdot \left({B}^{2} \cdot F\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(C, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left(F \cdot C\right), A\right)\right), \left(8 \cdot \left({B}^{2} \cdot F\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(C, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(F, C\right), A\right)\right), \left(8 \cdot \left({B}^{2} \cdot F\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(C, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(F, C\right), A\right)\right), \mathsf{*.f64}\left(8, \left({B}^{2} \cdot F\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(C, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(F, C\right), A\right)\right), \mathsf{*.f64}\left(8, \left(F \cdot {B}^{2}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(C, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(F, C\right), A\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(F, \left({B}^{2}\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(C, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(F, C\right), A\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(F, \left(B \cdot B\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       12. *-lowering-*.f64 26.8%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(C, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(F, C\right), A\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(F, \mathsf{*.f64}\left(B, B\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   12. Simplified 26.8%

       \[\leadsto \frac{\sqrt{\color{blue}{C \cdot \left(-16 \cdot \left(\left(F \cdot C\right) \cdot A\right) + 8 \cdot \left(F \cdot \left(B \cdot B\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if 9.50000000000000023e-304 < F < 7.50000000000000014e55

   1. Initial program 22.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   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. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(C, \left(\sqrt{{C}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(C, \left(\sqrt{C \cdot C + {B}^{2}}\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(C, \left(\sqrt{C \cdot C + B \cdot B}\right)\right)\right)\right)\right) \]

      12. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(C, \left(\mathsf{hypot}\left(C, B\right)\right)\right)\right)\right)\right) \]

      13. hypot-lowering-hypot.f64 21.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(C, B\right)\right)\right)\right)\right) \]

   5. Simplified 21.8%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)}} \]

   6. Taylor expanded in C around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \color{blue}{\left(\sqrt{B \cdot F}\right)}\right) \]
   7. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(B \cdot F\right)\right)\right) \]

      2. *-lowering-*.f64 16.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(B, F\right)\right)\right) \]

   8. Simplified 16.5%

      \[\leadsto \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{B \cdot F}} \]

   if 7.50000000000000014e55 < F

   1. Initial program 17.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{F}{B}}\right), \left(\sqrt{2}\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{F}{B}\right)\right), \left(\sqrt{2}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \left(\sqrt{2}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 22.3%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   5. Simplified 22.3%

      \[\leadsto \color{blue}{-\sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 20.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 9.5 \cdot 10^{-304}:\\ \;\;\;\;\frac{\sqrt{C \cdot \left(-16 \cdot \left(A \cdot \left(C \cdot F\right)\right) + \left(F \cdot \left(B \cdot B\right)\right) \cdot 8\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;F \leq 7.5 \cdot 10^{+55}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(0 - \sqrt{B \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(0 - \sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 34.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 8 \cdot 10^{-65}:\\ \;\;\;\;\frac{\sqrt{C \cdot \left(-16 \cdot \left(A \cdot \left(C \cdot F\right)\right) + \left(F \cdot \left(B\_m \cdot B\_m\right)\right) \cdot 8\right)}}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B\_m}} \cdot \left(0 - \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 8e-65)
   (/
    (sqrt (* C (+ (* -16.0 (* A (* C F))) (* (* F (* B_m B_m)) 8.0))))
    (- (* (* 4.0 A) C) (* B_m B_m)))
   (* (sqrt (/ F B_m)) (- 0.0 (sqrt 2.0)))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 8e-65) {
		tmp = sqrt((C * ((-16.0 * (A * (C * F))) + ((F * (B_m * B_m)) * 8.0)))) / (((4.0 * A) * C) - (B_m * B_m));
	} else {
		tmp = sqrt((F / B_m)) * (0.0 - sqrt(2.0));
	}
	return tmp;
}

Fortran

B_m = abs(b)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b_m <= 8d-65) then
        tmp = sqrt((c * (((-16.0d0) * (a * (c * f))) + ((f * (b_m * b_m)) * 8.0d0)))) / (((4.0d0 * a) * c) - (b_m * b_m))
    else
        tmp = sqrt((f / b_m)) * (0.0d0 - sqrt(2.0d0))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 8e-65) {
		tmp = Math.sqrt((C * ((-16.0 * (A * (C * F))) + ((F * (B_m * B_m)) * 8.0)))) / (((4.0 * A) * C) - (B_m * B_m));
	} else {
		tmp = Math.sqrt((F / B_m)) * (0.0 - Math.sqrt(2.0));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 8e-65:
		tmp = math.sqrt((C * ((-16.0 * (A * (C * F))) + ((F * (B_m * B_m)) * 8.0)))) / (((4.0 * A) * C) - (B_m * B_m))
	else:
		tmp = math.sqrt((F / B_m)) * (0.0 - math.sqrt(2.0))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 8e-65)
		tmp = Float64(sqrt(Float64(C * Float64(Float64(-16.0 * Float64(A * Float64(C * F))) + Float64(Float64(F * Float64(B_m * B_m)) * 8.0)))) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m)));
	else
		tmp = Float64(sqrt(Float64(F / B_m)) * Float64(0.0 - sqrt(2.0)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 8e-65)
		tmp = sqrt((C * ((-16.0 * (A * (C * F))) + ((F * (B_m * B_m)) * 8.0)))) / (((4.0 * A) * C) - (B_m * B_m));
	else
		tmp = sqrt((F / B_m)) * (0.0 - sqrt(2.0));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 8e-65], N[(N[Sqrt[N[(C * N[(N[(-16.0 * N[(A * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(F * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision] * N[(0.0 - N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 7.99999999999999939e-65

   1. Initial program 22.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 27.5%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right) \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. pow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right) \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right) \cdot \left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right) \cdot \left(2 \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right) \cdot \left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right), \left(2 \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   6. Applied egg-rr 24.5%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(\left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right) \cdot \left(2 \cdot F\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   7. Taylor expanded in A around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(-1 \cdot \left(A \cdot \left(-4 \cdot \frac{{B}^{2} \cdot C}{A} + 8 \cdot {C}^{2}\right)\right)\right)}, \mathsf{*.f64}\left(2, F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\mathsf{neg}\left(A \cdot \left(-4 \cdot \frac{{B}^{2} \cdot C}{A} + 8 \cdot {C}^{2}\right)\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(\left(A \cdot \left(-4 \cdot \frac{{B}^{2} \cdot C}{A} + 8 \cdot {C}^{2}\right)\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(A, \left(-4 \cdot \frac{{B}^{2} \cdot C}{A} + 8 \cdot {C}^{2}\right)\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(A, \mathsf{+.f64}\left(\left(-4 \cdot \frac{{B}^{2} \cdot C}{A}\right), \left(8 \cdot {C}^{2}\right)\right)\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(A, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-4, \left(\frac{{B}^{2} \cdot C}{A}\right)\right), \left(8 \cdot {C}^{2}\right)\right)\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(A, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left({B}^{2} \cdot C\right), A\right)\right), \left(8 \cdot {C}^{2}\right)\right)\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(A, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({B}^{2}\right), C\right), A\right)\right), \left(8 \cdot {C}^{2}\right)\right)\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(A, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(B \cdot B\right), C\right), A\right)\right), \left(8 \cdot {C}^{2}\right)\right)\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(A, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), C\right), A\right)\right), \left(8 \cdot {C}^{2}\right)\right)\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(A, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), C\right), A\right)\right), \mathsf{*.f64}\left(8, \left({C}^{2}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(A, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), C\right), A\right)\right), \mathsf{*.f64}\left(8, \left(C \cdot C\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      12. *-lowering-*.f64 11.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(A, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), C\right), A\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(C, C\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Simplified 11.8%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-A \cdot \left(-4 \cdot \frac{\left(B \cdot B\right) \cdot C}{A} + 8 \cdot \left(C \cdot C\right)\right)\right)} \cdot \left(2 \cdot F\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   10. Taylor expanded in C around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(C \cdot \left(-16 \cdot \left(A \cdot \left(C \cdot F\right)\right) + 8 \cdot \left({B}^{2} \cdot F\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   11. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(C, \left(-16 \cdot \left(A \cdot \left(C \cdot F\right)\right) + 8 \cdot \left({B}^{2} \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(C, \mathsf{+.f64}\left(\left(-16 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right), \left(8 \cdot \left({B}^{2} \cdot F\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(C, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \left(A \cdot \left(C \cdot F\right)\right)\right), \left(8 \cdot \left({B}^{2} \cdot F\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(C, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \left(\left(C \cdot F\right) \cdot A\right)\right), \left(8 \cdot \left({B}^{2} \cdot F\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(C, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left(C \cdot F\right), A\right)\right), \left(8 \cdot \left({B}^{2} \cdot F\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(C, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left(F \cdot C\right), A\right)\right), \left(8 \cdot \left({B}^{2} \cdot F\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(C, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(F, C\right), A\right)\right), \left(8 \cdot \left({B}^{2} \cdot F\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(C, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(F, C\right), A\right)\right), \mathsf{*.f64}\left(8, \left({B}^{2} \cdot F\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(C, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(F, C\right), A\right)\right), \mathsf{*.f64}\left(8, \left(F \cdot {B}^{2}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(C, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(F, C\right), A\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(F, \left({B}^{2}\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(C, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(F, C\right), A\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(F, \left(B \cdot B\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       12. *-lowering-*.f64 15.6%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(C, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(F, C\right), A\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(F, \mathsf{*.f64}\left(B, B\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   12. Simplified 15.6%

       \[\leadsto \frac{\sqrt{\color{blue}{C \cdot \left(-16 \cdot \left(\left(F \cdot C\right) \cdot A\right) + 8 \cdot \left(F \cdot \left(B \cdot B\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if 7.99999999999999939e-65 < B

   1. Initial program 22.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{F}{B}}\right), \left(\sqrt{2}\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{F}{B}\right)\right), \left(\sqrt{2}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \left(\sqrt{2}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 46.5%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   5. Simplified 46.5%

      \[\leadsto \color{blue}{-\sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 24.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 8 \cdot 10^{-65}:\\ \;\;\;\;\frac{\sqrt{C \cdot \left(-16 \cdot \left(A \cdot \left(C \cdot F\right)\right) + \left(F \cdot \left(B \cdot B\right)\right) \cdot 8\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(0 - \sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 17.7% accurate, 4.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m\\ \mathbf{if}\;C \leq -29000000000:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(F \cdot \left(A \cdot \left(A \cdot C\right)\right)\right)}}{t\_0}\\ \mathbf{elif}\;C \leq 62:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B\_m \cdot \left(B\_m \cdot B\_m\right)\right)\right)}}{t\_0}\\ \mathbf{elif}\;C \leq 1.1 \cdot 10^{+125}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \left(C \cdot C\right)\right) \cdot \left(A \cdot -16\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{B\_m}{2 \cdot {\left(C \cdot F\right)}^{0.5}}}\\ \end{array} \end{array} \]

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 (<= C -29000000000.0)
     (/ (sqrt (* -16.0 (* F (* A (* A C))))) t_0)
     (if (<= C 62.0)
       (/ (sqrt (* 2.0 (* F (* B_m (* B_m B_m))))) t_0)
       (if (<= C 1.1e+125)
         (/ (sqrt (* (* F (* C C)) (* A -16.0))) t_0)
         (/ -1.0 (/ B_m (* 2.0 (pow (* C F) 0.5)))))))))

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 (C <= -29000000000.0) {
		tmp = sqrt((-16.0 * (F * (A * (A * C))))) / t_0;
	} else if (C <= 62.0) {
		tmp = sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0;
	} else if (C <= 1.1e+125) {
		tmp = sqrt(((F * (C * C)) * (A * -16.0))) / t_0;
	} else {
		tmp = -1.0 / (B_m / (2.0 * pow((C * F), 0.5)));
	}
	return tmp;
}

Fortran

B_m = abs(b)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((4.0d0 * a) * c) - (b_m * b_m)
    if (c <= (-29000000000.0d0)) then
        tmp = sqrt(((-16.0d0) * (f * (a * (a * c))))) / t_0
    else if (c <= 62.0d0) then
        tmp = sqrt((2.0d0 * (f * (b_m * (b_m * b_m))))) / t_0
    else if (c <= 1.1d+125) then
        tmp = sqrt(((f * (c * c)) * (a * (-16.0d0)))) / t_0
    else
        tmp = (-1.0d0) / (b_m / (2.0d0 * ((c * f) ** 0.5d0)))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double t_0 = ((4.0 * A) * C) - (B_m * B_m);
	double tmp;
	if (C <= -29000000000.0) {
		tmp = Math.sqrt((-16.0 * (F * (A * (A * C))))) / t_0;
	} else if (C <= 62.0) {
		tmp = Math.sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0;
	} else if (C <= 1.1e+125) {
		tmp = Math.sqrt(((F * (C * C)) * (A * -16.0))) / t_0;
	} else {
		tmp = -1.0 / (B_m / (2.0 * Math.pow((C * F), 0.5)));
	}
	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 C <= -29000000000.0:
		tmp = math.sqrt((-16.0 * (F * (A * (A * C))))) / t_0
	elif C <= 62.0:
		tmp = math.sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0
	elif C <= 1.1e+125:
		tmp = math.sqrt(((F * (C * C)) * (A * -16.0))) / t_0
	else:
		tmp = -1.0 / (B_m / (2.0 * math.pow((C * F), 0.5)))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m))
	tmp = 0.0
	if (C <= -29000000000.0)
		tmp = Float64(sqrt(Float64(-16.0 * Float64(F * Float64(A * Float64(A * C))))) / t_0);
	elseif (C <= 62.0)
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(B_m * Float64(B_m * B_m))))) / t_0);
	elseif (C <= 1.1e+125)
		tmp = Float64(sqrt(Float64(Float64(F * Float64(C * C)) * Float64(A * -16.0))) / t_0);
	else
		tmp = Float64(-1.0 / Float64(B_m / Float64(2.0 * (Float64(C * F) ^ 0.5))));
	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 (C <= -29000000000.0)
		tmp = sqrt((-16.0 * (F * (A * (A * C))))) / t_0;
	elseif (C <= 62.0)
		tmp = sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0;
	elseif (C <= 1.1e+125)
		tmp = sqrt(((F * (C * C)) * (A * -16.0))) / t_0;
	else
		tmp = -1.0 / (B_m / (2.0 * ((C * F) ^ 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[C, -29000000000.0], N[(N[Sqrt[N[(-16.0 * N[(F * N[(A * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[C, 62.0], N[(N[Sqrt[N[(2.0 * N[(F * N[(B$95$m * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[C, 1.1e+125], N[(N[Sqrt[N[(N[(F * N[(C * C), $MachinePrecision]), $MachinePrecision] * N[(A * -16.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], N[(-1.0 / N[(B$95$m / N[(2.0 * N[Power[N[(C * F), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if C < -2.9e10

   1. Initial program 4.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 4.9%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left({A}^{2} \cdot \left(C \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left({A}^{2} \cdot C\right) \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left({A}^{2} \cdot C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({A}^{2}\right), C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(A \cdot A\right), C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 20.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 20.2%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
   8. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left(A \cdot \left(A \cdot C\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left(\left(A \cdot C\right) \cdot A\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(A \cdot C\right), A\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(C \cdot A\right), A\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 22.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(C, A\right), A\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Applied egg-rr 22.0%

      \[\leadsto \frac{\sqrt{-16 \cdot \left(\color{blue}{\left(\left(C \cdot A\right) \cdot A\right)} \cdot F\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if -2.9e10 < C < 62

   1. Initial program 26.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 31.5%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in B around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left({B}^{3} \cdot F\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left({B}^{3} \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left({B}^{3}\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(B \cdot \left(B \cdot B\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(B \cdot {B}^{2}\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \left({B}^{2}\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \left(B \cdot B\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 7.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 7.6%

      \[\leadsto \frac{\sqrt{\color{blue}{2 \cdot \left(\left(B \cdot \left(B \cdot B\right)\right) \cdot F\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if 62 < C < 1.09999999999999995e125

   1. Initial program 52.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 52.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in A around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(-16 \cdot A\right) \cdot \left({C}^{2} \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(-16 \cdot A\right), \left({C}^{2} \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, A\right), \left({C}^{2} \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, A\right), \mathsf{*.f64}\left(\left({C}^{2}\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, A\right), \mathsf{*.f64}\left(\left(C \cdot C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 43.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, A\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(C, C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 43.0%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-16 \cdot A\right) \cdot \left(\left(C \cdot C\right) \cdot F\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if 1.09999999999999995e125 < C

   1. Initial program 5.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(C, \left(\sqrt{{C}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(C, \left(\sqrt{C \cdot C + {B}^{2}}\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(C, \left(\sqrt{C \cdot C + B \cdot B}\right)\right)\right)\right)\right) \]

      12. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(C, \left(\mathsf{hypot}\left(C, B\right)\right)\right)\right)\right)\right) \]

      13. hypot-lowering-hypot.f64 13.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(C, B\right)\right)\right)\right)\right) \]

   5. Simplified 13.9%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{F \cdot \left(C + \sqrt{C \cdot C + B \cdot B}\right)} \cdot \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right)} \]

      2. pow1/2 N/A

         \[\leadsto {\left(F \cdot \left(C + \sqrt{C \cdot C + B \cdot B}\right)\right)}^{\frac{1}{2}} \cdot \left(\color{blue}{-1} \cdot \frac{\sqrt{2}}{B}\right) \]

      3. unpow-prod-down N/A

         \[\leadsto \left({F}^{\frac{1}{2}} \cdot {\left(C + \sqrt{C \cdot C + B \cdot B}\right)}^{\frac{1}{2}}\right) \cdot \left(\color{blue}{-1} \cdot \frac{\sqrt{2}}{B}\right) \]

      4. associate-*l* N/A

         \[\leadsto {F}^{\frac{1}{2}} \cdot \color{blue}{\left({\left(C + \sqrt{C \cdot C + B \cdot B}\right)}^{\frac{1}{2}} \cdot \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({F}^{\frac{1}{2}}\right), \color{blue}{\left({\left(C + \sqrt{C \cdot C + B \cdot B}\right)}^{\frac{1}{2}} \cdot \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right)}\right) \]

      6. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{F}\right), \left(\color{blue}{{\left(C + \sqrt{C \cdot C + B \cdot B}\right)}^{\frac{1}{2}}} \cdot \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \left(\color{blue}{{\left(C + \sqrt{C \cdot C + B \cdot B}\right)}^{\frac{1}{2}}} \cdot \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\left({\left(C + \sqrt{C \cdot C + B \cdot B}\right)}^{\frac{1}{2}}\right), \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right)}\right)\right) \]

      9. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\left(\sqrt{C + \sqrt{C \cdot C + B \cdot B}}\right), \left(\color{blue}{-1} \cdot \frac{\sqrt{2}}{B}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(C + \sqrt{C \cdot C + B \cdot B}\right)\right), \left(\color{blue}{-1} \cdot \frac{\sqrt{2}}{B}\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \left(\sqrt{C \cdot C + B \cdot B}\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \left(\sqrt{B \cdot B + C \cdot C}\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right)\right) \]

      13. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \left(\mathsf{hypot}\left(B, C\right)\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right)\right) \]

      14. hypot-lowering-hypot.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right)\right) \]

      15. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right), \left(\mathsf{neg}\left(\frac{\sqrt{2}}{B}\right)\right)\right)\right) \]

      16. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right), \left(0 - \color{blue}{\frac{\sqrt{2}}{B}}\right)\right)\right) \]

      17. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right), \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{\sqrt{2}}{B}\right)}\right)\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left(\sqrt{2}\right), \color{blue}{B}\right)\right)\right)\right) \]

   7. Applied egg-rr 17.0%

      \[\leadsto \color{blue}{\sqrt{F} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \left(0 - \frac{\sqrt{2}}{B}\right)\right)} \]

   8. Taylor expanded in C around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{{\left(\sqrt{2}\right)}^{2}}{B} \cdot \sqrt{C \cdot F}\right)} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{{\left(\sqrt{2}\right)}^{2}}{B} \cdot \sqrt{C \cdot F}\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{{\left(\sqrt{2}\right)}^{2}}{B} \cdot \sqrt{C \cdot F}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\frac{{\left(\sqrt{2}\right)}^{2}}{B}\right), \left(\sqrt{C \cdot F}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\frac{\sqrt{2} \cdot \sqrt{2}}{B}\right), \left(\sqrt{C \cdot F}\right)\right)\right) \]

      5. rem-square-sqrt N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\frac{2}{B}\right), \left(\sqrt{C \cdot F}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, B\right), \left(\sqrt{C \cdot F}\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, B\right), \mathsf{sqrt.f64}\left(\left(C \cdot F\right)\right)\right)\right) \]

      8. *-lowering-*.f64 11.3%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(C, F\right)\right)\right)\right) \]

   10. Simplified 11.3%

       \[\leadsto \color{blue}{-\frac{2}{B} \cdot \sqrt{C \cdot F}} \]
   11. Step-by-step derivation

       1. associate-*l/ N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{2 \cdot \sqrt{C \cdot F}}{B}\right)\right) \]

       2. clear-num N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{1}{\frac{B}{2 \cdot \sqrt{C \cdot F}}}\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{B}{2 \cdot \sqrt{C \cdot F}}\right)\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(B, \left(2 \cdot \sqrt{C \cdot F}\right)\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(B, \mathsf{*.f64}\left(2, \left(\sqrt{C \cdot F}\right)\right)\right)\right)\right) \]

       6. pow1/2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(B, \mathsf{*.f64}\left(2, \left({\left(C \cdot F\right)}^{\frac{1}{2}}\right)\right)\right)\right)\right) \]

       7. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(B, \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\left(C \cdot F\right), \frac{1}{2}\right)\right)\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(B, \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\left(F \cdot C\right), \frac{1}{2}\right)\right)\right)\right)\right) \]

       9. *-lowering-*.f64 11.7%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(B, \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(F, C\right), \frac{1}{2}\right)\right)\right)\right)\right) \]

   12. Applied egg-rr 11.7%

       \[\leadsto -\color{blue}{\frac{1}{\frac{B}{2 \cdot {\left(F \cdot C\right)}^{0.5}}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 15.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -29000000000:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(F \cdot \left(A \cdot \left(A \cdot C\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;C \leq 62:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot \left(B \cdot B\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;C \leq 1.1 \cdot 10^{+125}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \left(C \cdot C\right)\right) \cdot \left(A \cdot -16\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{B}{2 \cdot {\left(C \cdot F\right)}^{0.5}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 30.6% accurate, 4.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 5.5 \cdot 10^{-70}:\\ \;\;\;\;\frac{\sqrt{C \cdot \left(-16 \cdot \left(A \cdot \left(C \cdot F\right)\right) + \left(F \cdot \left(B\_m \cdot B\_m\right)\right) \cdot 8\right)}}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;0 - \sqrt{\frac{2 \cdot \left(F + \frac{C \cdot F}{B\_m}\right) + \frac{F \cdot \left(C \cdot C\right)}{B\_m \cdot B\_m}}{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 5.5e-70)
   (/
    (sqrt (* C (+ (* -16.0 (* A (* C F))) (* (* F (* B_m B_m)) 8.0))))
    (- (* (* 4.0 A) C) (* B_m B_m)))
   (-
    0.0
    (sqrt
     (/
      (+ (* 2.0 (+ F (/ (* C F) B_m))) (/ (* F (* C C)) (* B_m 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 <= 5.5e-70) {
		tmp = sqrt((C * ((-16.0 * (A * (C * F))) + ((F * (B_m * B_m)) * 8.0)))) / (((4.0 * A) * C) - (B_m * B_m));
	} else {
		tmp = 0.0 - sqrt((((2.0 * (F + ((C * F) / B_m))) + ((F * (C * C)) / (B_m * 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 <= 5.5d-70) then
        tmp = sqrt((c * (((-16.0d0) * (a * (c * f))) + ((f * (b_m * b_m)) * 8.0d0)))) / (((4.0d0 * a) * c) - (b_m * b_m))
    else
        tmp = 0.0d0 - sqrt((((2.0d0 * (f + ((c * f) / b_m))) + ((f * (c * c)) / (b_m * 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 <= 5.5e-70) {
		tmp = Math.sqrt((C * ((-16.0 * (A * (C * F))) + ((F * (B_m * B_m)) * 8.0)))) / (((4.0 * A) * C) - (B_m * B_m));
	} else {
		tmp = 0.0 - Math.sqrt((((2.0 * (F + ((C * F) / B_m))) + ((F * (C * C)) / (B_m * B_m))) / B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 5.5e-70:
		tmp = math.sqrt((C * ((-16.0 * (A * (C * F))) + ((F * (B_m * B_m)) * 8.0)))) / (((4.0 * A) * C) - (B_m * B_m))
	else:
		tmp = 0.0 - math.sqrt((((2.0 * (F + ((C * F) / B_m))) + ((F * (C * C)) / (B_m * B_m))) / B_m))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 5.5e-70)
		tmp = Float64(sqrt(Float64(C * Float64(Float64(-16.0 * Float64(A * Float64(C * F))) + Float64(Float64(F * Float64(B_m * B_m)) * 8.0)))) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m)));
	else
		tmp = Float64(0.0 - sqrt(Float64(Float64(Float64(2.0 * Float64(F + Float64(Float64(C * F) / B_m))) + Float64(Float64(F * Float64(C * C)) / Float64(B_m * 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 <= 5.5e-70)
		tmp = sqrt((C * ((-16.0 * (A * (C * F))) + ((F * (B_m * B_m)) * 8.0)))) / (((4.0 * A) * C) - (B_m * B_m));
	else
		tmp = 0.0 - sqrt((((2.0 * (F + ((C * F) / B_m))) + ((F * (C * C)) / (B_m * 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, 5.5e-70], N[(N[Sqrt[N[(C * N[(N[(-16.0 * N[(A * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(F * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[Sqrt[N[(N[(N[(2.0 * N[(F + N[(N[(C * F), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(F * N[(C * C), $MachinePrecision]), $MachinePrecision] / N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 5.5000000000000001e-70

   1. Initial program 22.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 27.5%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right) \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. pow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right) \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right) \cdot \left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right) \cdot \left(2 \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right) \cdot \left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right), \left(2 \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   6. Applied egg-rr 24.5%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(\left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right) \cdot \left(2 \cdot F\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   7. Taylor expanded in A around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(-1 \cdot \left(A \cdot \left(-4 \cdot \frac{{B}^{2} \cdot C}{A} + 8 \cdot {C}^{2}\right)\right)\right)}, \mathsf{*.f64}\left(2, F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\mathsf{neg}\left(A \cdot \left(-4 \cdot \frac{{B}^{2} \cdot C}{A} + 8 \cdot {C}^{2}\right)\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(\left(A \cdot \left(-4 \cdot \frac{{B}^{2} \cdot C}{A} + 8 \cdot {C}^{2}\right)\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(A, \left(-4 \cdot \frac{{B}^{2} \cdot C}{A} + 8 \cdot {C}^{2}\right)\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(A, \mathsf{+.f64}\left(\left(-4 \cdot \frac{{B}^{2} \cdot C}{A}\right), \left(8 \cdot {C}^{2}\right)\right)\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(A, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-4, \left(\frac{{B}^{2} \cdot C}{A}\right)\right), \left(8 \cdot {C}^{2}\right)\right)\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(A, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left({B}^{2} \cdot C\right), A\right)\right), \left(8 \cdot {C}^{2}\right)\right)\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(A, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({B}^{2}\right), C\right), A\right)\right), \left(8 \cdot {C}^{2}\right)\right)\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(A, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(B \cdot B\right), C\right), A\right)\right), \left(8 \cdot {C}^{2}\right)\right)\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(A, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), C\right), A\right)\right), \left(8 \cdot {C}^{2}\right)\right)\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(A, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), C\right), A\right)\right), \mathsf{*.f64}\left(8, \left({C}^{2}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(A, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), C\right), A\right)\right), \mathsf{*.f64}\left(8, \left(C \cdot C\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      12. *-lowering-*.f64 11.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(A, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), C\right), A\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(C, C\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Simplified 11.8%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-A \cdot \left(-4 \cdot \frac{\left(B \cdot B\right) \cdot C}{A} + 8 \cdot \left(C \cdot C\right)\right)\right)} \cdot \left(2 \cdot F\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   10. Taylor expanded in C around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(C \cdot \left(-16 \cdot \left(A \cdot \left(C \cdot F\right)\right) + 8 \cdot \left({B}^{2} \cdot F\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   11. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(C, \left(-16 \cdot \left(A \cdot \left(C \cdot F\right)\right) + 8 \cdot \left({B}^{2} \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(C, \mathsf{+.f64}\left(\left(-16 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right), \left(8 \cdot \left({B}^{2} \cdot F\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(C, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \left(A \cdot \left(C \cdot F\right)\right)\right), \left(8 \cdot \left({B}^{2} \cdot F\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(C, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \left(\left(C \cdot F\right) \cdot A\right)\right), \left(8 \cdot \left({B}^{2} \cdot F\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(C, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left(C \cdot F\right), A\right)\right), \left(8 \cdot \left({B}^{2} \cdot F\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(C, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left(F \cdot C\right), A\right)\right), \left(8 \cdot \left({B}^{2} \cdot F\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(C, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(F, C\right), A\right)\right), \left(8 \cdot \left({B}^{2} \cdot F\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(C, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(F, C\right), A\right)\right), \mathsf{*.f64}\left(8, \left({B}^{2} \cdot F\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(C, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(F, C\right), A\right)\right), \mathsf{*.f64}\left(8, \left(F \cdot {B}^{2}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(C, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(F, C\right), A\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(F, \left({B}^{2}\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(C, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(F, C\right), A\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(F, \left(B \cdot B\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       12. *-lowering-*.f64 15.6%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(C, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(F, C\right), A\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(F, \mathsf{*.f64}\left(B, B\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   12. Simplified 15.6%

       \[\leadsto \frac{\sqrt{\color{blue}{C \cdot \left(-16 \cdot \left(\left(F \cdot C\right) \cdot A\right) + 8 \cdot \left(F \cdot \left(B \cdot B\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if 5.5000000000000001e-70 < B

   1. Initial program 22.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 23.5%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in B around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left({B}^{3} \cdot \left(2 \cdot F + \left(2 \cdot \frac{F \cdot \left(A + C\right)}{B} + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left({B}^{3}\right), \left(2 \cdot F + \left(2 \cdot \frac{F \cdot \left(A + C\right)}{B} + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(B \cdot \left(B \cdot B\right)\right), \left(2 \cdot F + \left(2 \cdot \frac{F \cdot \left(A + C\right)}{B} + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(B \cdot {B}^{2}\right), \left(2 \cdot F + \left(2 \cdot \frac{F \cdot \left(A + C\right)}{B} + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \left({B}^{2}\right)\right), \left(2 \cdot F + \left(2 \cdot \frac{F \cdot \left(A + C\right)}{B} + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \left(B \cdot B\right)\right), \left(2 \cdot F + \left(2 \cdot \frac{F \cdot \left(A + C\right)}{B} + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \left(2 \cdot F + \left(2 \cdot \frac{F \cdot \left(A + C\right)}{B} + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \left(\left(2 \cdot F + 2 \cdot \frac{F \cdot \left(A + C\right)}{B}\right) + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{+.f64}\left(\left(2 \cdot F + 2 \cdot \frac{F \cdot \left(A + C\right)}{B}\right), \left(2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 12.9%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(B \cdot \left(B \cdot B\right)\right) \cdot \left(2 \cdot \left(F + \frac{F \cdot \left(C + A\right)}{B}\right) + \frac{\left(2 \cdot F\right) \cdot \left(-4 \cdot \left(A \cdot C\right) + 0.5 \cdot \left(\left(A - C\right) \cdot \left(A - C\right)\right)\right)}{B \cdot B}\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   8. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{2 \cdot \left(F + \frac{C \cdot F}{B}\right) + \frac{{C}^{2} \cdot F}{{B}^{2}}}{B}}} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{2 \cdot \left(F + \frac{C \cdot F}{B}\right) + \frac{{C}^{2} \cdot F}{{B}^{2}}}{B}}\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{\frac{2 \cdot \left(F + \frac{C \cdot F}{B}\right) + \frac{{C}^{2} \cdot F}{{B}^{2}}}{B}}\right)\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{2 \cdot \left(F + \frac{C \cdot F}{B}\right) + \frac{{C}^{2} \cdot F}{{B}^{2}}}{B}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \left(F + \frac{C \cdot F}{B}\right) + \frac{{C}^{2} \cdot F}{{B}^{2}}\right), B\right)\right)\right) \]

   10. Simplified 40.7%

       \[\leadsto \color{blue}{-\sqrt{\frac{2 \cdot \left(F + \frac{C \cdot F}{B}\right) + \frac{\left(C \cdot C\right) \cdot F}{B \cdot B}}{B}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 22.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 5.5 \cdot 10^{-70}:\\ \;\;\;\;\frac{\sqrt{C \cdot \left(-16 \cdot \left(A \cdot \left(C \cdot F\right)\right) + \left(F \cdot \left(B \cdot B\right)\right) \cdot 8\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;0 - \sqrt{\frac{2 \cdot \left(F + \frac{C \cdot F}{B}\right) + \frac{F \cdot \left(C \cdot C\right)}{B \cdot B}}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 17.0% accurate, 4.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m\\ \mathbf{if}\;B\_m \leq 3 \cdot 10^{-95}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(F \cdot \left(A \cdot \left(A \cdot C\right)\right)\right)}}{t\_0}\\ \mathbf{elif}\;B\_m \leq 6 \cdot 10^{+102}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B\_m \cdot \left(B\_m \cdot B\_m\right)\right)\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{B\_m}{2 \cdot {\left(C \cdot F\right)}^{0.5}}}\\ \end{array} \end{array} \]

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 3e-95)
     (/ (sqrt (* -16.0 (* F (* A (* A C))))) t_0)
     (if (<= B_m 6e+102)
       (/ (sqrt (* 2.0 (* F (* B_m (* B_m B_m))))) t_0)
       (/ -1.0 (/ B_m (* 2.0 (pow (* C F) 0.5))))))))

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 <= 3e-95) {
		tmp = sqrt((-16.0 * (F * (A * (A * C))))) / t_0;
	} else if (B_m <= 6e+102) {
		tmp = sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0;
	} else {
		tmp = -1.0 / (B_m / (2.0 * pow((C * F), 0.5)));
	}
	return tmp;
}

Fortran

B_m = abs(b)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((4.0d0 * a) * c) - (b_m * b_m)
    if (b_m <= 3d-95) then
        tmp = sqrt(((-16.0d0) * (f * (a * (a * c))))) / t_0
    else if (b_m <= 6d+102) then
        tmp = sqrt((2.0d0 * (f * (b_m * (b_m * b_m))))) / t_0
    else
        tmp = (-1.0d0) / (b_m / (2.0d0 * ((c * f) ** 0.5d0)))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double t_0 = ((4.0 * A) * C) - (B_m * B_m);
	double tmp;
	if (B_m <= 3e-95) {
		tmp = Math.sqrt((-16.0 * (F * (A * (A * C))))) / t_0;
	} else if (B_m <= 6e+102) {
		tmp = Math.sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0;
	} else {
		tmp = -1.0 / (B_m / (2.0 * Math.pow((C * F), 0.5)));
	}
	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 <= 3e-95:
		tmp = math.sqrt((-16.0 * (F * (A * (A * C))))) / t_0
	elif B_m <= 6e+102:
		tmp = math.sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0
	else:
		tmp = -1.0 / (B_m / (2.0 * math.pow((C * F), 0.5)))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m))
	tmp = 0.0
	if (B_m <= 3e-95)
		tmp = Float64(sqrt(Float64(-16.0 * Float64(F * Float64(A * Float64(A * C))))) / t_0);
	elseif (B_m <= 6e+102)
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(B_m * Float64(B_m * B_m))))) / t_0);
	else
		tmp = Float64(-1.0 / Float64(B_m / Float64(2.0 * (Float64(C * F) ^ 0.5))));
	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 <= 3e-95)
		tmp = sqrt((-16.0 * (F * (A * (A * C))))) / t_0;
	elseif (B_m <= 6e+102)
		tmp = sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0;
	else
		tmp = -1.0 / (B_m / (2.0 * ((C * F) ^ 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 3e-95], N[(N[Sqrt[N[(-16.0 * N[(F * N[(A * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[B$95$m, 6e+102], N[(N[Sqrt[N[(2.0 * N[(F * N[(B$95$m * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], N[(-1.0 / N[(B$95$m / N[(2.0 * N[Power[N[(C * F), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 3e-95

   1. Initial program 21.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 27.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left({A}^{2} \cdot \left(C \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left({A}^{2} \cdot C\right) \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left({A}^{2} \cdot C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({A}^{2}\right), C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(A \cdot A\right), C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 10.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 10.2%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
   8. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left(A \cdot \left(A \cdot C\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left(\left(A \cdot C\right) \cdot A\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(A \cdot C\right), A\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(C \cdot A\right), A\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 12.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(C, A\right), A\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Applied egg-rr 12.9%

      \[\leadsto \frac{\sqrt{-16 \cdot \left(\color{blue}{\left(\left(C \cdot A\right) \cdot A\right)} \cdot F\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if 3e-95 < B < 5.9999999999999996e102

   1. Initial program 48.3%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 51.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in B around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left({B}^{3} \cdot F\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left({B}^{3} \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left({B}^{3}\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(B \cdot \left(B \cdot B\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(B \cdot {B}^{2}\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \left({B}^{2}\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \left(B \cdot B\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 27.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 27.0%

      \[\leadsto \frac{\sqrt{\color{blue}{2 \cdot \left(\left(B \cdot \left(B \cdot B\right)\right) \cdot F\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if 5.9999999999999996e102 < B

   1. Initial program 3.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in 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. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(C, \left(\sqrt{{C}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(C, \left(\sqrt{C \cdot C + {B}^{2}}\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(C, \left(\sqrt{C \cdot C + B \cdot B}\right)\right)\right)\right)\right) \]

      12. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(C, \left(\mathsf{hypot}\left(C, B\right)\right)\right)\right)\right)\right) \]

      13. hypot-lowering-hypot.f64 41.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(C, B\right)\right)\right)\right)\right) \]

   5. Simplified 41.7%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{F \cdot \left(C + \sqrt{C \cdot C + B \cdot B}\right)} \cdot \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right)} \]

      2. pow1/2 N/A

         \[\leadsto {\left(F \cdot \left(C + \sqrt{C \cdot C + B \cdot B}\right)\right)}^{\frac{1}{2}} \cdot \left(\color{blue}{-1} \cdot \frac{\sqrt{2}}{B}\right) \]

      3. unpow-prod-down N/A

         \[\leadsto \left({F}^{\frac{1}{2}} \cdot {\left(C + \sqrt{C \cdot C + B \cdot B}\right)}^{\frac{1}{2}}\right) \cdot \left(\color{blue}{-1} \cdot \frac{\sqrt{2}}{B}\right) \]

      4. associate-*l* N/A

         \[\leadsto {F}^{\frac{1}{2}} \cdot \color{blue}{\left({\left(C + \sqrt{C \cdot C + B \cdot B}\right)}^{\frac{1}{2}} \cdot \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({F}^{\frac{1}{2}}\right), \color{blue}{\left({\left(C + \sqrt{C \cdot C + B \cdot B}\right)}^{\frac{1}{2}} \cdot \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right)}\right) \]

      6. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{F}\right), \left(\color{blue}{{\left(C + \sqrt{C \cdot C + B \cdot B}\right)}^{\frac{1}{2}}} \cdot \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \left(\color{blue}{{\left(C + \sqrt{C \cdot C + B \cdot B}\right)}^{\frac{1}{2}}} \cdot \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\left({\left(C + \sqrt{C \cdot C + B \cdot B}\right)}^{\frac{1}{2}}\right), \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right)}\right)\right) \]

      9. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\left(\sqrt{C + \sqrt{C \cdot C + B \cdot B}}\right), \left(\color{blue}{-1} \cdot \frac{\sqrt{2}}{B}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(C + \sqrt{C \cdot C + B \cdot B}\right)\right), \left(\color{blue}{-1} \cdot \frac{\sqrt{2}}{B}\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \left(\sqrt{C \cdot C + B \cdot B}\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \left(\sqrt{B \cdot B + C \cdot C}\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right)\right) \]

      13. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \left(\mathsf{hypot}\left(B, C\right)\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right)\right) \]

      14. hypot-lowering-hypot.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right)\right) \]

      15. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right), \left(\mathsf{neg}\left(\frac{\sqrt{2}}{B}\right)\right)\right)\right) \]

      16. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right), \left(0 - \color{blue}{\frac{\sqrt{2}}{B}}\right)\right)\right) \]

      17. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right), \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{\sqrt{2}}{B}\right)}\right)\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left(\sqrt{2}\right), \color{blue}{B}\right)\right)\right)\right) \]

   7. Applied egg-rr 72.8%

      \[\leadsto \color{blue}{\sqrt{F} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \left(0 - \frac{\sqrt{2}}{B}\right)\right)} \]

   8. Taylor expanded in C around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{{\left(\sqrt{2}\right)}^{2}}{B} \cdot \sqrt{C \cdot F}\right)} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{{\left(\sqrt{2}\right)}^{2}}{B} \cdot \sqrt{C \cdot F}\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{{\left(\sqrt{2}\right)}^{2}}{B} \cdot \sqrt{C \cdot F}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\frac{{\left(\sqrt{2}\right)}^{2}}{B}\right), \left(\sqrt{C \cdot F}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\frac{\sqrt{2} \cdot \sqrt{2}}{B}\right), \left(\sqrt{C \cdot F}\right)\right)\right) \]

      5. rem-square-sqrt N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\frac{2}{B}\right), \left(\sqrt{C \cdot F}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, B\right), \left(\sqrt{C \cdot F}\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, B\right), \mathsf{sqrt.f64}\left(\left(C \cdot F\right)\right)\right)\right) \]

      8. *-lowering-*.f64 9.6%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(C, F\right)\right)\right)\right) \]

   10. Simplified 9.6%

       \[\leadsto \color{blue}{-\frac{2}{B} \cdot \sqrt{C \cdot F}} \]
   11. Step-by-step derivation

       1. associate-*l/ N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{2 \cdot \sqrt{C \cdot F}}{B}\right)\right) \]

       2. clear-num N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{1}{\frac{B}{2 \cdot \sqrt{C \cdot F}}}\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{B}{2 \cdot \sqrt{C \cdot F}}\right)\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(B, \left(2 \cdot \sqrt{C \cdot F}\right)\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(B, \mathsf{*.f64}\left(2, \left(\sqrt{C \cdot F}\right)\right)\right)\right)\right) \]

       6. pow1/2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(B, \mathsf{*.f64}\left(2, \left({\left(C \cdot F\right)}^{\frac{1}{2}}\right)\right)\right)\right)\right) \]

       7. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(B, \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\left(C \cdot F\right), \frac{1}{2}\right)\right)\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(B, \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\left(F \cdot C\right), \frac{1}{2}\right)\right)\right)\right)\right) \]

       9. *-lowering-*.f64 9.8%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(B, \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(F, C\right), \frac{1}{2}\right)\right)\right)\right)\right) \]

   12. Applied egg-rr 9.8%

       \[\leadsto -\color{blue}{\frac{1}{\frac{B}{2 \cdot {\left(F \cdot C\right)}^{0.5}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 14.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3 \cdot 10^{-95}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(F \cdot \left(A \cdot \left(A \cdot C\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;B \leq 6 \cdot 10^{+102}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot \left(B \cdot B\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{B}{2 \cdot {\left(C \cdot F\right)}^{0.5}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 27.9% accurate, 5.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 8.2 \cdot 10^{-55}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(-8 \cdot \left(A \cdot \left(C \cdot C\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;0 - \sqrt{\frac{2 \cdot \left(F + \frac{C \cdot F}{B\_m}\right) + \frac{F \cdot \left(C \cdot C\right)}{B\_m \cdot B\_m}}{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 8.2e-55)
   (/
    (sqrt (* (* 2.0 F) (* -8.0 (* A (* C C)))))
    (- (* (* 4.0 A) C) (* B_m B_m)))
   (-
    0.0
    (sqrt
     (/
      (+ (* 2.0 (+ F (/ (* C F) B_m))) (/ (* F (* C C)) (* B_m 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 <= 8.2e-55) {
		tmp = sqrt(((2.0 * F) * (-8.0 * (A * (C * C))))) / (((4.0 * A) * C) - (B_m * B_m));
	} else {
		tmp = 0.0 - sqrt((((2.0 * (F + ((C * F) / B_m))) + ((F * (C * C)) / (B_m * 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 <= 8.2d-55) then
        tmp = sqrt(((2.0d0 * f) * ((-8.0d0) * (a * (c * c))))) / (((4.0d0 * a) * c) - (b_m * b_m))
    else
        tmp = 0.0d0 - sqrt((((2.0d0 * (f + ((c * f) / b_m))) + ((f * (c * c)) / (b_m * 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 <= 8.2e-55) {
		tmp = Math.sqrt(((2.0 * F) * (-8.0 * (A * (C * C))))) / (((4.0 * A) * C) - (B_m * B_m));
	} else {
		tmp = 0.0 - Math.sqrt((((2.0 * (F + ((C * F) / B_m))) + ((F * (C * C)) / (B_m * B_m))) / B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 8.2e-55:
		tmp = math.sqrt(((2.0 * F) * (-8.0 * (A * (C * C))))) / (((4.0 * A) * C) - (B_m * B_m))
	else:
		tmp = 0.0 - math.sqrt((((2.0 * (F + ((C * F) / B_m))) + ((F * (C * C)) / (B_m * B_m))) / B_m))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 8.2e-55)
		tmp = Float64(sqrt(Float64(Float64(2.0 * F) * Float64(-8.0 * Float64(A * Float64(C * C))))) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m)));
	else
		tmp = Float64(0.0 - sqrt(Float64(Float64(Float64(2.0 * Float64(F + Float64(Float64(C * F) / B_m))) + Float64(Float64(F * Float64(C * C)) / Float64(B_m * 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 <= 8.2e-55)
		tmp = sqrt(((2.0 * F) * (-8.0 * (A * (C * C))))) / (((4.0 * A) * C) - (B_m * B_m));
	else
		tmp = 0.0 - sqrt((((2.0 * (F + ((C * F) / B_m))) + ((F * (C * C)) / (B_m * 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, 8.2e-55], N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(-8.0 * N[(A * N[(C * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[Sqrt[N[(N[(N[(2.0 * N[(F + N[(N[(C * F), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(F * N[(C * C), $MachinePrecision]), $MachinePrecision] / N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 8.1999999999999996e-55

   1. Initial program 22.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 28.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right) \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. pow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right) \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right) \cdot \left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right) \cdot \left(2 \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right) \cdot \left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right), \left(2 \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   6. Applied egg-rr 25.2%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(\left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right) \cdot \left(2 \cdot F\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   7. Taylor expanded in A around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(-8 \cdot \left(A \cdot {C}^{2}\right)\right)}, \mathsf{*.f64}\left(2, F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-8, \left(A \cdot {C}^{2}\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-8, \mathsf{*.f64}\left(A, \left({C}^{2}\right)\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-8, \mathsf{*.f64}\left(A, \left(C \cdot C\right)\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 12.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-8, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, C\right)\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Simplified 12.6%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-8 \cdot \left(A \cdot \left(C \cdot C\right)\right)\right)} \cdot \left(2 \cdot F\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if 8.1999999999999996e-55 < B

   1. Initial program 20.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 21.6%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in B around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left({B}^{3} \cdot \left(2 \cdot F + \left(2 \cdot \frac{F \cdot \left(A + C\right)}{B} + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left({B}^{3}\right), \left(2 \cdot F + \left(2 \cdot \frac{F \cdot \left(A + C\right)}{B} + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(B \cdot \left(B \cdot B\right)\right), \left(2 \cdot F + \left(2 \cdot \frac{F \cdot \left(A + C\right)}{B} + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(B \cdot {B}^{2}\right), \left(2 \cdot F + \left(2 \cdot \frac{F \cdot \left(A + C\right)}{B} + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \left({B}^{2}\right)\right), \left(2 \cdot F + \left(2 \cdot \frac{F \cdot \left(A + C\right)}{B} + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \left(B \cdot B\right)\right), \left(2 \cdot F + \left(2 \cdot \frac{F \cdot \left(A + C\right)}{B} + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \left(2 \cdot F + \left(2 \cdot \frac{F \cdot \left(A + C\right)}{B} + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \left(\left(2 \cdot F + 2 \cdot \frac{F \cdot \left(A + C\right)}{B}\right) + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{+.f64}\left(\left(2 \cdot F + 2 \cdot \frac{F \cdot \left(A + C\right)}{B}\right), \left(2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 11.8%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(B \cdot \left(B \cdot B\right)\right) \cdot \left(2 \cdot \left(F + \frac{F \cdot \left(C + A\right)}{B}\right) + \frac{\left(2 \cdot F\right) \cdot \left(-4 \cdot \left(A \cdot C\right) + 0.5 \cdot \left(\left(A - C\right) \cdot \left(A - C\right)\right)\right)}{B \cdot B}\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   8. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{2 \cdot \left(F + \frac{C \cdot F}{B}\right) + \frac{{C}^{2} \cdot F}{{B}^{2}}}{B}}} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{2 \cdot \left(F + \frac{C \cdot F}{B}\right) + \frac{{C}^{2} \cdot F}{{B}^{2}}}{B}}\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{\frac{2 \cdot \left(F + \frac{C \cdot F}{B}\right) + \frac{{C}^{2} \cdot F}{{B}^{2}}}{B}}\right)\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{2 \cdot \left(F + \frac{C \cdot F}{B}\right) + \frac{{C}^{2} \cdot F}{{B}^{2}}}{B}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \left(F + \frac{C \cdot F}{B}\right) + \frac{{C}^{2} \cdot F}{{B}^{2}}\right), B\right)\right)\right) \]

   10. Simplified 40.9%

       \[\leadsto \color{blue}{-\sqrt{\frac{2 \cdot \left(F + \frac{C \cdot F}{B}\right) + \frac{\left(C \cdot C\right) \cdot F}{B \cdot B}}{B}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 20.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 8.2 \cdot 10^{-55}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(-8 \cdot \left(A \cdot \left(C \cdot C\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;0 - \sqrt{\frac{2 \cdot \left(F + \frac{C \cdot F}{B}\right) + \frac{F \cdot \left(C \cdot C\right)}{B \cdot B}}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 28.1% accurate, 5.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 1.65 \cdot 10^{-56}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(-8 \cdot \left(A \cdot \left(C \cdot C\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;0 - \sqrt{\frac{2 \cdot \left(F + \frac{A \cdot F}{B\_m}\right) + \frac{F \cdot \left(A \cdot A\right)}{B\_m \cdot B\_m}}{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 1.65e-56)
   (/
    (sqrt (* (* 2.0 F) (* -8.0 (* A (* C C)))))
    (- (* (* 4.0 A) C) (* B_m B_m)))
   (-
    0.0
    (sqrt
     (/
      (+ (* 2.0 (+ F (/ (* A F) B_m))) (/ (* F (* A A)) (* B_m 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 <= 1.65e-56) {
		tmp = sqrt(((2.0 * F) * (-8.0 * (A * (C * C))))) / (((4.0 * A) * C) - (B_m * B_m));
	} else {
		tmp = 0.0 - sqrt((((2.0 * (F + ((A * F) / B_m))) + ((F * (A * A)) / (B_m * 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 <= 1.65d-56) then
        tmp = sqrt(((2.0d0 * f) * ((-8.0d0) * (a * (c * c))))) / (((4.0d0 * a) * c) - (b_m * b_m))
    else
        tmp = 0.0d0 - sqrt((((2.0d0 * (f + ((a * f) / b_m))) + ((f * (a * a)) / (b_m * 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 <= 1.65e-56) {
		tmp = Math.sqrt(((2.0 * F) * (-8.0 * (A * (C * C))))) / (((4.0 * A) * C) - (B_m * B_m));
	} else {
		tmp = 0.0 - Math.sqrt((((2.0 * (F + ((A * F) / B_m))) + ((F * (A * A)) / (B_m * B_m))) / B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 1.65e-56:
		tmp = math.sqrt(((2.0 * F) * (-8.0 * (A * (C * C))))) / (((4.0 * A) * C) - (B_m * B_m))
	else:
		tmp = 0.0 - math.sqrt((((2.0 * (F + ((A * F) / B_m))) + ((F * (A * A)) / (B_m * B_m))) / B_m))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 1.65e-56)
		tmp = Float64(sqrt(Float64(Float64(2.0 * F) * Float64(-8.0 * Float64(A * Float64(C * C))))) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m)));
	else
		tmp = Float64(0.0 - sqrt(Float64(Float64(Float64(2.0 * Float64(F + Float64(Float64(A * F) / B_m))) + Float64(Float64(F * Float64(A * A)) / Float64(B_m * 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 <= 1.65e-56)
		tmp = sqrt(((2.0 * F) * (-8.0 * (A * (C * C))))) / (((4.0 * A) * C) - (B_m * B_m));
	else
		tmp = 0.0 - sqrt((((2.0 * (F + ((A * F) / B_m))) + ((F * (A * A)) / (B_m * 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, 1.65e-56], N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(-8.0 * N[(A * N[(C * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[Sqrt[N[(N[(N[(2.0 * N[(F + N[(N[(A * F), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(F * N[(A * A), $MachinePrecision]), $MachinePrecision] / N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1.64999999999999992e-56

   1. Initial program 22.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 28.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right) \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. pow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right) \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right) \cdot \left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right) \cdot \left(2 \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right) \cdot \left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right), \left(2 \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   6. Applied egg-rr 25.2%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(\left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right) \cdot \left(2 \cdot F\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   7. Taylor expanded in A around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(-8 \cdot \left(A \cdot {C}^{2}\right)\right)}, \mathsf{*.f64}\left(2, F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-8, \left(A \cdot {C}^{2}\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-8, \mathsf{*.f64}\left(A, \left({C}^{2}\right)\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-8, \mathsf{*.f64}\left(A, \left(C \cdot C\right)\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 12.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-8, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, C\right)\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Simplified 12.6%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-8 \cdot \left(A \cdot \left(C \cdot C\right)\right)\right)} \cdot \left(2 \cdot F\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if 1.64999999999999992e-56 < B

   1. Initial program 20.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 21.6%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in B around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left({B}^{3} \cdot \left(2 \cdot F + \left(2 \cdot \frac{F \cdot \left(A + C\right)}{B} + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left({B}^{3}\right), \left(2 \cdot F + \left(2 \cdot \frac{F \cdot \left(A + C\right)}{B} + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(B \cdot \left(B \cdot B\right)\right), \left(2 \cdot F + \left(2 \cdot \frac{F \cdot \left(A + C\right)}{B} + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(B \cdot {B}^{2}\right), \left(2 \cdot F + \left(2 \cdot \frac{F \cdot \left(A + C\right)}{B} + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \left({B}^{2}\right)\right), \left(2 \cdot F + \left(2 \cdot \frac{F \cdot \left(A + C\right)}{B} + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \left(B \cdot B\right)\right), \left(2 \cdot F + \left(2 \cdot \frac{F \cdot \left(A + C\right)}{B} + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \left(2 \cdot F + \left(2 \cdot \frac{F \cdot \left(A + C\right)}{B} + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \left(\left(2 \cdot F + 2 \cdot \frac{F \cdot \left(A + C\right)}{B}\right) + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{+.f64}\left(\left(2 \cdot F + 2 \cdot \frac{F \cdot \left(A + C\right)}{B}\right), \left(2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 11.8%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(B \cdot \left(B \cdot B\right)\right) \cdot \left(2 \cdot \left(F + \frac{F \cdot \left(C + A\right)}{B}\right) + \frac{\left(2 \cdot F\right) \cdot \left(-4 \cdot \left(A \cdot C\right) + 0.5 \cdot \left(\left(A - C\right) \cdot \left(A - C\right)\right)\right)}{B \cdot B}\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   8. Taylor expanded in C around 0

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{2 \cdot \left(F + \frac{A \cdot F}{B}\right) + \frac{{A}^{2} \cdot F}{{B}^{2}}}{B}}} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{2 \cdot \left(F + \frac{A \cdot F}{B}\right) + \frac{{A}^{2} \cdot F}{{B}^{2}}}{B}}\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{\frac{2 \cdot \left(F + \frac{A \cdot F}{B}\right) + \frac{{A}^{2} \cdot F}{{B}^{2}}}{B}}\right)\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{2 \cdot \left(F + \frac{A \cdot F}{B}\right) + \frac{{A}^{2} \cdot F}{{B}^{2}}}{B}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \left(F + \frac{A \cdot F}{B}\right) + \frac{{A}^{2} \cdot F}{{B}^{2}}\right), B\right)\right)\right) \]

   10. Simplified 36.9%

       \[\leadsto \color{blue}{-\sqrt{\frac{2 \cdot \left(F + \frac{A \cdot F}{B}\right) + \frac{\left(A \cdot A\right) \cdot F}{B \cdot B}}{B}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 19.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.65 \cdot 10^{-56}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(-8 \cdot \left(A \cdot \left(C \cdot C\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;0 - \sqrt{\frac{2 \cdot \left(F + \frac{A \cdot F}{B}\right) + \frac{F \cdot \left(A \cdot A\right)}{B \cdot B}}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 14.0% accurate, 5.1× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;C \leq 4.4 \cdot 10^{-61}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(F \cdot \left(A \cdot \left(A \cdot C\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot {\left(C \cdot F\right)}^{0.5}}{0 - B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= C 4.4e-61)
   (/ (sqrt (* -16.0 (* F (* A (* A C))))) (- (* (* 4.0 A) C) (* B_m B_m)))
   (/ (* 2.0 (pow (* C F) 0.5)) (- 0.0 B_m))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= 4.4e-61) {
		tmp = sqrt((-16.0 * (F * (A * (A * C))))) / (((4.0 * A) * C) - (B_m * B_m));
	} else {
		tmp = (2.0 * pow((C * F), 0.5)) / (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 (c <= 4.4d-61) then
        tmp = sqrt(((-16.0d0) * (f * (a * (a * c))))) / (((4.0d0 * a) * c) - (b_m * b_m))
    else
        tmp = (2.0d0 * ((c * f) ** 0.5d0)) / (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 (C <= 4.4e-61) {
		tmp = Math.sqrt((-16.0 * (F * (A * (A * C))))) / (((4.0 * A) * C) - (B_m * B_m));
	} else {
		tmp = (2.0 * Math.pow((C * F), 0.5)) / (0.0 - B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if C <= 4.4e-61:
		tmp = math.sqrt((-16.0 * (F * (A * (A * C))))) / (((4.0 * A) * C) - (B_m * B_m))
	else:
		tmp = (2.0 * math.pow((C * F), 0.5)) / (0.0 - B_m)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (C <= 4.4e-61)
		tmp = Float64(sqrt(Float64(-16.0 * Float64(F * Float64(A * Float64(A * C))))) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m)));
	else
		tmp = Float64(Float64(2.0 * (Float64(C * F) ^ 0.5)) / 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 (C <= 4.4e-61)
		tmp = sqrt((-16.0 * (F * (A * (A * C))))) / (((4.0 * A) * C) - (B_m * B_m));
	else
		tmp = (2.0 * ((C * F) ^ 0.5)) / (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[C, 4.4e-61], N[(N[Sqrt[N[(-16.0 * N[(F * N[(A * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Power[N[(C * F), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < 4.40000000000000017e-61

   1. Initial program 19.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 23.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left({A}^{2} \cdot \left(C \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left({A}^{2} \cdot C\right) \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left({A}^{2} \cdot C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({A}^{2}\right), C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(A \cdot A\right), C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 11.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 11.1%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
   8. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left(A \cdot \left(A \cdot C\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left(\left(A \cdot C\right) \cdot A\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(A \cdot C\right), A\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(C \cdot A\right), A\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 14.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(C, A\right), A\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Applied egg-rr 14.0%

      \[\leadsto \frac{\sqrt{-16 \cdot \left(\color{blue}{\left(\left(C \cdot A\right) \cdot A\right)} \cdot F\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if 4.40000000000000017e-61 < C

   1. Initial program 29.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in 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. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(C, \left(\sqrt{{C}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(C, \left(\sqrt{C \cdot C + {B}^{2}}\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(C, \left(\sqrt{C \cdot C + B \cdot B}\right)\right)\right)\right)\right) \]

      12. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(C, \left(\mathsf{hypot}\left(C, B\right)\right)\right)\right)\right)\right) \]

      13. hypot-lowering-hypot.f64 15.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(C, B\right)\right)\right)\right)\right) \]

   5. Simplified 15.4%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{F \cdot \left(C + \sqrt{C \cdot C + B \cdot B}\right)} \cdot \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right)} \]

      2. pow1/2 N/A

         \[\leadsto {\left(F \cdot \left(C + \sqrt{C \cdot C + B \cdot B}\right)\right)}^{\frac{1}{2}} \cdot \left(\color{blue}{-1} \cdot \frac{\sqrt{2}}{B}\right) \]

      3. unpow-prod-down N/A

         \[\leadsto \left({F}^{\frac{1}{2}} \cdot {\left(C + \sqrt{C \cdot C + B \cdot B}\right)}^{\frac{1}{2}}\right) \cdot \left(\color{blue}{-1} \cdot \frac{\sqrt{2}}{B}\right) \]

      4. associate-*l* N/A

         \[\leadsto {F}^{\frac{1}{2}} \cdot \color{blue}{\left({\left(C + \sqrt{C \cdot C + B \cdot B}\right)}^{\frac{1}{2}} \cdot \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({F}^{\frac{1}{2}}\right), \color{blue}{\left({\left(C + \sqrt{C \cdot C + B \cdot B}\right)}^{\frac{1}{2}} \cdot \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right)}\right) \]

      6. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{F}\right), \left(\color{blue}{{\left(C + \sqrt{C \cdot C + B \cdot B}\right)}^{\frac{1}{2}}} \cdot \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \left(\color{blue}{{\left(C + \sqrt{C \cdot C + B \cdot B}\right)}^{\frac{1}{2}}} \cdot \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\left({\left(C + \sqrt{C \cdot C + B \cdot B}\right)}^{\frac{1}{2}}\right), \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right)}\right)\right) \]

      9. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\left(\sqrt{C + \sqrt{C \cdot C + B \cdot B}}\right), \left(\color{blue}{-1} \cdot \frac{\sqrt{2}}{B}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(C + \sqrt{C \cdot C + B \cdot B}\right)\right), \left(\color{blue}{-1} \cdot \frac{\sqrt{2}}{B}\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \left(\sqrt{C \cdot C + B \cdot B}\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \left(\sqrt{B \cdot B + C \cdot C}\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right)\right) \]

      13. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \left(\mathsf{hypot}\left(B, C\right)\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right)\right) \]

      14. hypot-lowering-hypot.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right)\right) \]

      15. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right), \left(\mathsf{neg}\left(\frac{\sqrt{2}}{B}\right)\right)\right)\right) \]

      16. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right), \left(0 - \color{blue}{\frac{\sqrt{2}}{B}}\right)\right)\right) \]

      17. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right), \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{\sqrt{2}}{B}\right)}\right)\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left(\sqrt{2}\right), \color{blue}{B}\right)\right)\right)\right) \]

   7. Applied egg-rr 17.5%

      \[\leadsto \color{blue}{\sqrt{F} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \left(0 - \frac{\sqrt{2}}{B}\right)\right)} \]

   8. Taylor expanded in C around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{{\left(\sqrt{2}\right)}^{2}}{B} \cdot \sqrt{C \cdot F}\right)} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{{\left(\sqrt{2}\right)}^{2}}{B} \cdot \sqrt{C \cdot F}\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{{\left(\sqrt{2}\right)}^{2}}{B} \cdot \sqrt{C \cdot F}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\frac{{\left(\sqrt{2}\right)}^{2}}{B}\right), \left(\sqrt{C \cdot F}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\frac{\sqrt{2} \cdot \sqrt{2}}{B}\right), \left(\sqrt{C \cdot F}\right)\right)\right) \]

      5. rem-square-sqrt N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\frac{2}{B}\right), \left(\sqrt{C \cdot F}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, B\right), \left(\sqrt{C \cdot F}\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, B\right), \mathsf{sqrt.f64}\left(\left(C \cdot F\right)\right)\right)\right) \]

      8. *-lowering-*.f64 10.0%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(C, F\right)\right)\right)\right) \]

   10. Simplified 10.0%

       \[\leadsto \color{blue}{-\frac{2}{B} \cdot \sqrt{C \cdot F}} \]
   11. Step-by-step derivation

       1. associate-*l/ N/A

          \[\leadsto \mathsf{neg}\left(\frac{2 \cdot \sqrt{C \cdot F}}{B}\right) \]

       2. distribute-neg-frac N/A

          \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \sqrt{C \cdot F}\right)}{\color{blue}{B}} \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(2 \cdot \sqrt{C \cdot F}\right)\right), \color{blue}{B}\right) \]

       4. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\left(2 \cdot \sqrt{C \cdot F}\right)\right), B\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(2, \left(\sqrt{C \cdot F}\right)\right)\right), B\right) \]

       6. pow1/2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(2, \left({\left(C \cdot F\right)}^{\frac{1}{2}}\right)\right)\right), B\right) \]

       7. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\left(C \cdot F\right), \frac{1}{2}\right)\right)\right), B\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\left(F \cdot C\right), \frac{1}{2}\right)\right)\right), B\right) \]

       9. *-lowering-*.f64 10.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(F, C\right), \frac{1}{2}\right)\right)\right), B\right) \]

   12. Applied egg-rr 10.2%

       \[\leadsto \color{blue}{\frac{-2 \cdot {\left(F \cdot C\right)}^{0.5}}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 12.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 4.4 \cdot 10^{-61}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(F \cdot \left(A \cdot \left(A \cdot C\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot {\left(C \cdot F\right)}^{0.5}}{0 - B}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 15.1% accurate, 5.1× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;C \leq 9.6 \cdot 10^{-61}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(A \cdot \left(F \cdot \left(A \cdot C\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot {\left(C \cdot F\right)}^{0.5}}{0 - B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= C 9.6e-61)
   (/ (sqrt (* -16.0 (* A (* F (* A C))))) (- (* (* 4.0 A) C) (* B_m B_m)))
   (/ (* 2.0 (pow (* C F) 0.5)) (- 0.0 B_m))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= 9.6e-61) {
		tmp = sqrt((-16.0 * (A * (F * (A * C))))) / (((4.0 * A) * C) - (B_m * B_m));
	} else {
		tmp = (2.0 * pow((C * F), 0.5)) / (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 (c <= 9.6d-61) then
        tmp = sqrt(((-16.0d0) * (a * (f * (a * c))))) / (((4.0d0 * a) * c) - (b_m * b_m))
    else
        tmp = (2.0d0 * ((c * f) ** 0.5d0)) / (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 (C <= 9.6e-61) {
		tmp = Math.sqrt((-16.0 * (A * (F * (A * C))))) / (((4.0 * A) * C) - (B_m * B_m));
	} else {
		tmp = (2.0 * Math.pow((C * F), 0.5)) / (0.0 - B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if C <= 9.6e-61:
		tmp = math.sqrt((-16.0 * (A * (F * (A * C))))) / (((4.0 * A) * C) - (B_m * B_m))
	else:
		tmp = (2.0 * math.pow((C * F), 0.5)) / (0.0 - B_m)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (C <= 9.6e-61)
		tmp = Float64(sqrt(Float64(-16.0 * Float64(A * Float64(F * Float64(A * C))))) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m)));
	else
		tmp = Float64(Float64(2.0 * (Float64(C * F) ^ 0.5)) / 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 (C <= 9.6e-61)
		tmp = sqrt((-16.0 * (A * (F * (A * C))))) / (((4.0 * A) * C) - (B_m * B_m));
	else
		tmp = (2.0 * ((C * F) ^ 0.5)) / (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[C, 9.6e-61], N[(N[Sqrt[N[(-16.0 * N[(A * N[(F * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Power[N[(C * F), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < 9.6000000000000004e-61

   1. Initial program 19.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 23.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left({A}^{2} \cdot \left(C \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left({A}^{2} \cdot C\right) \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left({A}^{2} \cdot C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({A}^{2}\right), C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(A \cdot A\right), C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 11.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 11.1%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
   8. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left(A \cdot \left(A \cdot C\right)\right) \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(A \cdot \left(\left(A \cdot C\right) \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \left(\left(A \cdot C\right) \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\left(A \cdot C\right), F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\left(C \cdot A\right), F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 13.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\mathsf{*.f64}\left(C, A\right), F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Applied egg-rr 13.4%

      \[\leadsto \frac{\sqrt{-16 \cdot \color{blue}{\left(A \cdot \left(\left(C \cdot A\right) \cdot F\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if 9.6000000000000004e-61 < C

   1. Initial program 29.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in 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. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(C, \left(\sqrt{{C}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(C, \left(\sqrt{C \cdot C + {B}^{2}}\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(C, \left(\sqrt{C \cdot C + B \cdot B}\right)\right)\right)\right)\right) \]

      12. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(C, \left(\mathsf{hypot}\left(C, B\right)\right)\right)\right)\right)\right) \]

      13. hypot-lowering-hypot.f64 15.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(C, B\right)\right)\right)\right)\right) \]

   5. Simplified 15.4%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{F \cdot \left(C + \sqrt{C \cdot C + B \cdot B}\right)} \cdot \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right)} \]

      2. pow1/2 N/A

         \[\leadsto {\left(F \cdot \left(C + \sqrt{C \cdot C + B \cdot B}\right)\right)}^{\frac{1}{2}} \cdot \left(\color{blue}{-1} \cdot \frac{\sqrt{2}}{B}\right) \]

      3. unpow-prod-down N/A

         \[\leadsto \left({F}^{\frac{1}{2}} \cdot {\left(C + \sqrt{C \cdot C + B \cdot B}\right)}^{\frac{1}{2}}\right) \cdot \left(\color{blue}{-1} \cdot \frac{\sqrt{2}}{B}\right) \]

      4. associate-*l* N/A

         \[\leadsto {F}^{\frac{1}{2}} \cdot \color{blue}{\left({\left(C + \sqrt{C \cdot C + B \cdot B}\right)}^{\frac{1}{2}} \cdot \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({F}^{\frac{1}{2}}\right), \color{blue}{\left({\left(C + \sqrt{C \cdot C + B \cdot B}\right)}^{\frac{1}{2}} \cdot \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right)}\right) \]

      6. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{F}\right), \left(\color{blue}{{\left(C + \sqrt{C \cdot C + B \cdot B}\right)}^{\frac{1}{2}}} \cdot \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \left(\color{blue}{{\left(C + \sqrt{C \cdot C + B \cdot B}\right)}^{\frac{1}{2}}} \cdot \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\left({\left(C + \sqrt{C \cdot C + B \cdot B}\right)}^{\frac{1}{2}}\right), \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right)}\right)\right) \]

      9. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\left(\sqrt{C + \sqrt{C \cdot C + B \cdot B}}\right), \left(\color{blue}{-1} \cdot \frac{\sqrt{2}}{B}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(C + \sqrt{C \cdot C + B \cdot B}\right)\right), \left(\color{blue}{-1} \cdot \frac{\sqrt{2}}{B}\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \left(\sqrt{C \cdot C + B \cdot B}\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \left(\sqrt{B \cdot B + C \cdot C}\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right)\right) \]

      13. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \left(\mathsf{hypot}\left(B, C\right)\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right)\right) \]

      14. hypot-lowering-hypot.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right)\right) \]

      15. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right), \left(\mathsf{neg}\left(\frac{\sqrt{2}}{B}\right)\right)\right)\right) \]

      16. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right), \left(0 - \color{blue}{\frac{\sqrt{2}}{B}}\right)\right)\right) \]

      17. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right), \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{\sqrt{2}}{B}\right)}\right)\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left(\sqrt{2}\right), \color{blue}{B}\right)\right)\right)\right) \]

   7. Applied egg-rr 17.5%

      \[\leadsto \color{blue}{\sqrt{F} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \left(0 - \frac{\sqrt{2}}{B}\right)\right)} \]

   8. Taylor expanded in C around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{{\left(\sqrt{2}\right)}^{2}}{B} \cdot \sqrt{C \cdot F}\right)} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{{\left(\sqrt{2}\right)}^{2}}{B} \cdot \sqrt{C \cdot F}\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{{\left(\sqrt{2}\right)}^{2}}{B} \cdot \sqrt{C \cdot F}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\frac{{\left(\sqrt{2}\right)}^{2}}{B}\right), \left(\sqrt{C \cdot F}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\frac{\sqrt{2} \cdot \sqrt{2}}{B}\right), \left(\sqrt{C \cdot F}\right)\right)\right) \]

      5. rem-square-sqrt N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\frac{2}{B}\right), \left(\sqrt{C \cdot F}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, B\right), \left(\sqrt{C \cdot F}\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, B\right), \mathsf{sqrt.f64}\left(\left(C \cdot F\right)\right)\right)\right) \]

      8. *-lowering-*.f64 10.0%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(C, F\right)\right)\right)\right) \]

   10. Simplified 10.0%

       \[\leadsto \color{blue}{-\frac{2}{B} \cdot \sqrt{C \cdot F}} \]
   11. Step-by-step derivation

       1. associate-*l/ N/A

          \[\leadsto \mathsf{neg}\left(\frac{2 \cdot \sqrt{C \cdot F}}{B}\right) \]

       2. distribute-neg-frac N/A

          \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \sqrt{C \cdot F}\right)}{\color{blue}{B}} \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(2 \cdot \sqrt{C \cdot F}\right)\right), \color{blue}{B}\right) \]

       4. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\left(2 \cdot \sqrt{C \cdot F}\right)\right), B\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(2, \left(\sqrt{C \cdot F}\right)\right)\right), B\right) \]

       6. pow1/2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(2, \left({\left(C \cdot F\right)}^{\frac{1}{2}}\right)\right)\right), B\right) \]

       7. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\left(C \cdot F\right), \frac{1}{2}\right)\right)\right), B\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\left(F \cdot C\right), \frac{1}{2}\right)\right)\right), B\right) \]

       9. *-lowering-*.f64 10.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(F, C\right), \frac{1}{2}\right)\right)\right), B\right) \]

   12. Applied egg-rr 10.2%

       \[\leadsto \color{blue}{\frac{-2 \cdot {\left(F \cdot C\right)}^{0.5}}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 12.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 9.6 \cdot 10^{-61}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(A \cdot \left(F \cdot \left(A \cdot C\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot {\left(C \cdot F\right)}^{0.5}}{0 - B}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 7.1% accurate, 5.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;C \leq -1.05 \cdot 10^{-304}:\\ \;\;\;\;0.25 \cdot \left(\sqrt{B\_m \cdot F} \cdot \frac{1}{C}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot {\left(C \cdot F\right)}^{0.5}}{0 - B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= C -1.05e-304)
   (* 0.25 (* (sqrt (* B_m F)) (/ 1.0 C)))
   (/ (* 2.0 (pow (* C F) 0.5)) (- 0.0 B_m))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= -1.05e-304) {
		tmp = 0.25 * (sqrt((B_m * F)) * (1.0 / C));
	} else {
		tmp = (2.0 * pow((C * F), 0.5)) / (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 (c <= (-1.05d-304)) then
        tmp = 0.25d0 * (sqrt((b_m * f)) * (1.0d0 / c))
    else
        tmp = (2.0d0 * ((c * f) ** 0.5d0)) / (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 (C <= -1.05e-304) {
		tmp = 0.25 * (Math.sqrt((B_m * F)) * (1.0 / C));
	} else {
		tmp = (2.0 * Math.pow((C * F), 0.5)) / (0.0 - B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if C <= -1.05e-304:
		tmp = 0.25 * (math.sqrt((B_m * F)) * (1.0 / C))
	else:
		tmp = (2.0 * math.pow((C * F), 0.5)) / (0.0 - B_m)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (C <= -1.05e-304)
		tmp = Float64(0.25 * Float64(sqrt(Float64(B_m * F)) * Float64(1.0 / C)));
	else
		tmp = Float64(Float64(2.0 * (Float64(C * F) ^ 0.5)) / 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 (C <= -1.05e-304)
		tmp = 0.25 * (sqrt((B_m * F)) * (1.0 / C));
	else
		tmp = (2.0 * ((C * F) ^ 0.5)) / (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[C, -1.05e-304], N[(0.25 * N[(N[Sqrt[N[(B$95$m * F), $MachinePrecision]], $MachinePrecision] * N[(1.0 / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Power[N[(C * F), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < -1.05000000000000004e-304

   1. Initial program 16.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 21.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in B around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left({B}^{3} \cdot \left(2 \cdot F + \left(2 \cdot \frac{F \cdot \left(A + C\right)}{B} + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left({B}^{3}\right), \left(2 \cdot F + \left(2 \cdot \frac{F \cdot \left(A + C\right)}{B} + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(B \cdot \left(B \cdot B\right)\right), \left(2 \cdot F + \left(2 \cdot \frac{F \cdot \left(A + C\right)}{B} + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(B \cdot {B}^{2}\right), \left(2 \cdot F + \left(2 \cdot \frac{F \cdot \left(A + C\right)}{B} + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \left({B}^{2}\right)\right), \left(2 \cdot F + \left(2 \cdot \frac{F \cdot \left(A + C\right)}{B} + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \left(B \cdot B\right)\right), \left(2 \cdot F + \left(2 \cdot \frac{F \cdot \left(A + C\right)}{B} + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \left(2 \cdot F + \left(2 \cdot \frac{F \cdot \left(A + C\right)}{B} + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \left(\left(2 \cdot F + 2 \cdot \frac{F \cdot \left(A + C\right)}{B}\right) + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{+.f64}\left(\left(2 \cdot F + 2 \cdot \frac{F \cdot \left(A + C\right)}{B}\right), \left(2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 3.2%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(B \cdot \left(B \cdot B\right)\right) \cdot \left(2 \cdot \left(F + \frac{F \cdot \left(C + A\right)}{B}\right) + \frac{\left(2 \cdot F\right) \cdot \left(-4 \cdot \left(A \cdot C\right) + 0.5 \cdot \left(\left(A - C\right) \cdot \left(A - C\right)\right)\right)}{B \cdot B}\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   8. Taylor expanded in A around inf

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(\sqrt{B \cdot F} \cdot \frac{1}{C}\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \color{blue}{\left(\sqrt{B \cdot F} \cdot \frac{1}{C}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(\sqrt{B \cdot F}\right), \color{blue}{\left(\frac{1}{C}\right)}\right)\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(B \cdot F\right)\right), \left(\frac{\color{blue}{1}}{C}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(B, F\right)\right), \left(\frac{1}{C}\right)\right)\right) \]

      5. /-lowering-/.f64 2.4%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(B, F\right)\right), \mathsf{/.f64}\left(1, \color{blue}{C}\right)\right)\right) \]

   10. Simplified 2.4%

       \[\leadsto \color{blue}{0.25 \cdot \left(\sqrt{B \cdot F} \cdot \frac{1}{C}\right)} \]

   if -1.05000000000000004e-304 < C

   1. Initial program 26.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(C, \left(\sqrt{{C}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(C, \left(\sqrt{C \cdot C + {B}^{2}}\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(C, \left(\sqrt{C \cdot C + B \cdot B}\right)\right)\right)\right)\right) \]

      12. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(C, \left(\mathsf{hypot}\left(C, B\right)\right)\right)\right)\right)\right) \]

      13. hypot-lowering-hypot.f64 17.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(C, B\right)\right)\right)\right)\right) \]

   5. Simplified 17.7%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{F \cdot \left(C + \sqrt{C \cdot C + B \cdot B}\right)} \cdot \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right)} \]

      2. pow1/2 N/A

         \[\leadsto {\left(F \cdot \left(C + \sqrt{C \cdot C + B \cdot B}\right)\right)}^{\frac{1}{2}} \cdot \left(\color{blue}{-1} \cdot \frac{\sqrt{2}}{B}\right) \]

      3. unpow-prod-down N/A

         \[\leadsto \left({F}^{\frac{1}{2}} \cdot {\left(C + \sqrt{C \cdot C + B \cdot B}\right)}^{\frac{1}{2}}\right) \cdot \left(\color{blue}{-1} \cdot \frac{\sqrt{2}}{B}\right) \]

      4. associate-*l* N/A

         \[\leadsto {F}^{\frac{1}{2}} \cdot \color{blue}{\left({\left(C + \sqrt{C \cdot C + B \cdot B}\right)}^{\frac{1}{2}} \cdot \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({F}^{\frac{1}{2}}\right), \color{blue}{\left({\left(C + \sqrt{C \cdot C + B \cdot B}\right)}^{\frac{1}{2}} \cdot \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right)}\right) \]

      6. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{F}\right), \left(\color{blue}{{\left(C + \sqrt{C \cdot C + B \cdot B}\right)}^{\frac{1}{2}}} \cdot \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \left(\color{blue}{{\left(C + \sqrt{C \cdot C + B \cdot B}\right)}^{\frac{1}{2}}} \cdot \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\left({\left(C + \sqrt{C \cdot C + B \cdot B}\right)}^{\frac{1}{2}}\right), \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right)}\right)\right) \]

      9. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\left(\sqrt{C + \sqrt{C \cdot C + B \cdot B}}\right), \left(\color{blue}{-1} \cdot \frac{\sqrt{2}}{B}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(C + \sqrt{C \cdot C + B \cdot B}\right)\right), \left(\color{blue}{-1} \cdot \frac{\sqrt{2}}{B}\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \left(\sqrt{C \cdot C + B \cdot B}\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \left(\sqrt{B \cdot B + C \cdot C}\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right)\right) \]

      13. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \left(\mathsf{hypot}\left(B, C\right)\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right)\right) \]

      14. hypot-lowering-hypot.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right)\right) \]

      15. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right), \left(\mathsf{neg}\left(\frac{\sqrt{2}}{B}\right)\right)\right)\right) \]

      16. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right), \left(0 - \color{blue}{\frac{\sqrt{2}}{B}}\right)\right)\right) \]

      17. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right), \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{\sqrt{2}}{B}\right)}\right)\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left(\sqrt{2}\right), \color{blue}{B}\right)\right)\right)\right) \]

   7. Applied egg-rr 20.3%

      \[\leadsto \color{blue}{\sqrt{F} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \left(0 - \frac{\sqrt{2}}{B}\right)\right)} \]

   8. Taylor expanded in C around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{{\left(\sqrt{2}\right)}^{2}}{B} \cdot \sqrt{C \cdot F}\right)} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{{\left(\sqrt{2}\right)}^{2}}{B} \cdot \sqrt{C \cdot F}\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{{\left(\sqrt{2}\right)}^{2}}{B} \cdot \sqrt{C \cdot F}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\frac{{\left(\sqrt{2}\right)}^{2}}{B}\right), \left(\sqrt{C \cdot F}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\frac{\sqrt{2} \cdot \sqrt{2}}{B}\right), \left(\sqrt{C \cdot F}\right)\right)\right) \]

      5. rem-square-sqrt N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\frac{2}{B}\right), \left(\sqrt{C \cdot F}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, B\right), \left(\sqrt{C \cdot F}\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, B\right), \mathsf{sqrt.f64}\left(\left(C \cdot F\right)\right)\right)\right) \]

      8. *-lowering-*.f64 7.4%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(C, F\right)\right)\right)\right) \]

   10. Simplified 7.4%

       \[\leadsto \color{blue}{-\frac{2}{B} \cdot \sqrt{C \cdot F}} \]
   11. Step-by-step derivation

       1. associate-*l/ N/A

          \[\leadsto \mathsf{neg}\left(\frac{2 \cdot \sqrt{C \cdot F}}{B}\right) \]

       2. distribute-neg-frac N/A

          \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \sqrt{C \cdot F}\right)}{\color{blue}{B}} \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(2 \cdot \sqrt{C \cdot F}\right)\right), \color{blue}{B}\right) \]

       4. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\left(2 \cdot \sqrt{C \cdot F}\right)\right), B\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(2, \left(\sqrt{C \cdot F}\right)\right)\right), B\right) \]

       6. pow1/2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(2, \left({\left(C \cdot F\right)}^{\frac{1}{2}}\right)\right)\right), B\right) \]

       7. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\left(C \cdot F\right), \frac{1}{2}\right)\right)\right), B\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\left(F \cdot C\right), \frac{1}{2}\right)\right)\right), B\right) \]

       9. *-lowering-*.f64 7.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(F, C\right), \frac{1}{2}\right)\right)\right), B\right) \]

   12. Applied egg-rr 7.5%

       \[\leadsto \color{blue}{\frac{-2 \cdot {\left(F \cdot C\right)}^{0.5}}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 5.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -1.05 \cdot 10^{-304}:\\ \;\;\;\;0.25 \cdot \left(\sqrt{B \cdot F} \cdot \frac{1}{C}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot {\left(C \cdot F\right)}^{0.5}}{0 - B}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 7.1% accurate, 5.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;C \leq -1.3 \cdot 10^{-304}:\\ \;\;\;\;0.25 \cdot \left(\sqrt{B\_m \cdot F} \cdot \frac{1}{C}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(C \cdot F\right)}^{0.5} \cdot \frac{-2}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= C -1.3e-304)
   (* 0.25 (* (sqrt (* B_m F)) (/ 1.0 C)))
   (* (pow (* C F) 0.5) (/ -2.0 B_m))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= -1.3e-304) {
		tmp = 0.25 * (sqrt((B_m * F)) * (1.0 / C));
	} else {
		tmp = pow((C * F), 0.5) * (-2.0 / 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 (c <= (-1.3d-304)) then
        tmp = 0.25d0 * (sqrt((b_m * f)) * (1.0d0 / c))
    else
        tmp = ((c * f) ** 0.5d0) * ((-2.0d0) / 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 (C <= -1.3e-304) {
		tmp = 0.25 * (Math.sqrt((B_m * F)) * (1.0 / C));
	} else {
		tmp = Math.pow((C * F), 0.5) * (-2.0 / B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if C <= -1.3e-304:
		tmp = 0.25 * (math.sqrt((B_m * F)) * (1.0 / C))
	else:
		tmp = math.pow((C * F), 0.5) * (-2.0 / B_m)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (C <= -1.3e-304)
		tmp = Float64(0.25 * Float64(sqrt(Float64(B_m * F)) * Float64(1.0 / C)));
	else
		tmp = Float64((Float64(C * F) ^ 0.5) * Float64(-2.0 / B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (C <= -1.3e-304)
		tmp = 0.25 * (sqrt((B_m * F)) * (1.0 / C));
	else
		tmp = ((C * F) ^ 0.5) * (-2.0 / B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[C, -1.3e-304], N[(0.25 * N[(N[Sqrt[N[(B$95$m * F), $MachinePrecision]], $MachinePrecision] * N[(1.0 / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(C * F), $MachinePrecision], 0.5], $MachinePrecision] * N[(-2.0 / B$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < -1.29999999999999998e-304

   1. Initial program 16.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 21.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in B around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left({B}^{3} \cdot \left(2 \cdot F + \left(2 \cdot \frac{F \cdot \left(A + C\right)}{B} + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left({B}^{3}\right), \left(2 \cdot F + \left(2 \cdot \frac{F \cdot \left(A + C\right)}{B} + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(B \cdot \left(B \cdot B\right)\right), \left(2 \cdot F + \left(2 \cdot \frac{F \cdot \left(A + C\right)}{B} + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(B \cdot {B}^{2}\right), \left(2 \cdot F + \left(2 \cdot \frac{F \cdot \left(A + C\right)}{B} + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \left({B}^{2}\right)\right), \left(2 \cdot F + \left(2 \cdot \frac{F \cdot \left(A + C\right)}{B} + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \left(B \cdot B\right)\right), \left(2 \cdot F + \left(2 \cdot \frac{F \cdot \left(A + C\right)}{B} + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \left(2 \cdot F + \left(2 \cdot \frac{F \cdot \left(A + C\right)}{B} + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \left(\left(2 \cdot F + 2 \cdot \frac{F \cdot \left(A + C\right)}{B}\right) + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{+.f64}\left(\left(2 \cdot F + 2 \cdot \frac{F \cdot \left(A + C\right)}{B}\right), \left(2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 3.2%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(B \cdot \left(B \cdot B\right)\right) \cdot \left(2 \cdot \left(F + \frac{F \cdot \left(C + A\right)}{B}\right) + \frac{\left(2 \cdot F\right) \cdot \left(-4 \cdot \left(A \cdot C\right) + 0.5 \cdot \left(\left(A - C\right) \cdot \left(A - C\right)\right)\right)}{B \cdot B}\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   8. Taylor expanded in A around inf

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(\sqrt{B \cdot F} \cdot \frac{1}{C}\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \color{blue}{\left(\sqrt{B \cdot F} \cdot \frac{1}{C}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(\sqrt{B \cdot F}\right), \color{blue}{\left(\frac{1}{C}\right)}\right)\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(B \cdot F\right)\right), \left(\frac{\color{blue}{1}}{C}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(B, F\right)\right), \left(\frac{1}{C}\right)\right)\right) \]

      5. /-lowering-/.f64 2.4%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(B, F\right)\right), \mathsf{/.f64}\left(1, \color{blue}{C}\right)\right)\right) \]

   10. Simplified 2.4%

       \[\leadsto \color{blue}{0.25 \cdot \left(\sqrt{B \cdot F} \cdot \frac{1}{C}\right)} \]

   if -1.29999999999999998e-304 < C

   1. Initial program 26.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(C, \left(\sqrt{{C}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(C, \left(\sqrt{C \cdot C + {B}^{2}}\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(C, \left(\sqrt{C \cdot C + B \cdot B}\right)\right)\right)\right)\right) \]

      12. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(C, \left(\mathsf{hypot}\left(C, B\right)\right)\right)\right)\right)\right) \]

      13. hypot-lowering-hypot.f64 17.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(C, B\right)\right)\right)\right)\right) \]

   5. Simplified 17.7%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{F \cdot \left(C + \sqrt{C \cdot C + B \cdot B}\right)} \cdot \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right)} \]

      2. pow1/2 N/A

         \[\leadsto {\left(F \cdot \left(C + \sqrt{C \cdot C + B \cdot B}\right)\right)}^{\frac{1}{2}} \cdot \left(\color{blue}{-1} \cdot \frac{\sqrt{2}}{B}\right) \]

      3. unpow-prod-down N/A

         \[\leadsto \left({F}^{\frac{1}{2}} \cdot {\left(C + \sqrt{C \cdot C + B \cdot B}\right)}^{\frac{1}{2}}\right) \cdot \left(\color{blue}{-1} \cdot \frac{\sqrt{2}}{B}\right) \]

      4. associate-*l* N/A

         \[\leadsto {F}^{\frac{1}{2}} \cdot \color{blue}{\left({\left(C + \sqrt{C \cdot C + B \cdot B}\right)}^{\frac{1}{2}} \cdot \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({F}^{\frac{1}{2}}\right), \color{blue}{\left({\left(C + \sqrt{C \cdot C + B \cdot B}\right)}^{\frac{1}{2}} \cdot \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right)}\right) \]

      6. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{F}\right), \left(\color{blue}{{\left(C + \sqrt{C \cdot C + B \cdot B}\right)}^{\frac{1}{2}}} \cdot \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \left(\color{blue}{{\left(C + \sqrt{C \cdot C + B \cdot B}\right)}^{\frac{1}{2}}} \cdot \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\left({\left(C + \sqrt{C \cdot C + B \cdot B}\right)}^{\frac{1}{2}}\right), \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right)}\right)\right) \]

      9. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\left(\sqrt{C + \sqrt{C \cdot C + B \cdot B}}\right), \left(\color{blue}{-1} \cdot \frac{\sqrt{2}}{B}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(C + \sqrt{C \cdot C + B \cdot B}\right)\right), \left(\color{blue}{-1} \cdot \frac{\sqrt{2}}{B}\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \left(\sqrt{C \cdot C + B \cdot B}\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \left(\sqrt{B \cdot B + C \cdot C}\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right)\right) \]

      13. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \left(\mathsf{hypot}\left(B, C\right)\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right)\right) \]

      14. hypot-lowering-hypot.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right)\right) \]

      15. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right), \left(\mathsf{neg}\left(\frac{\sqrt{2}}{B}\right)\right)\right)\right) \]

      16. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right), \left(0 - \color{blue}{\frac{\sqrt{2}}{B}}\right)\right)\right) \]

      17. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right), \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{\sqrt{2}}{B}\right)}\right)\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left(\sqrt{2}\right), \color{blue}{B}\right)\right)\right)\right) \]

   7. Applied egg-rr 20.3%

      \[\leadsto \color{blue}{\sqrt{F} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \left(0 - \frac{\sqrt{2}}{B}\right)\right)} \]

   8. Taylor expanded in C around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{{\left(\sqrt{2}\right)}^{2}}{B} \cdot \sqrt{C \cdot F}\right)} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{{\left(\sqrt{2}\right)}^{2}}{B} \cdot \sqrt{C \cdot F}\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{{\left(\sqrt{2}\right)}^{2}}{B} \cdot \sqrt{C \cdot F}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\frac{{\left(\sqrt{2}\right)}^{2}}{B}\right), \left(\sqrt{C \cdot F}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\frac{\sqrt{2} \cdot \sqrt{2}}{B}\right), \left(\sqrt{C \cdot F}\right)\right)\right) \]

      5. rem-square-sqrt N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\frac{2}{B}\right), \left(\sqrt{C \cdot F}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, B\right), \left(\sqrt{C \cdot F}\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, B\right), \mathsf{sqrt.f64}\left(\left(C \cdot F\right)\right)\right)\right) \]

      8. *-lowering-*.f64 7.4%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(C, F\right)\right)\right)\right) \]

   10. Simplified 7.4%

       \[\leadsto \color{blue}{-\frac{2}{B} \cdot \sqrt{C \cdot F}} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{C \cdot F} \cdot \frac{2}{B}\right) \]

       2. distribute-rgt-neg-in N/A

          \[\leadsto \sqrt{C \cdot F} \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{2}{B}\right)\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{C \cdot F}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{2}{B}\right)\right)}\right) \]

       4. pow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({\left(C \cdot F\right)}^{\frac{1}{2}}\right), \left(\mathsf{neg}\left(\color{blue}{\frac{2}{B}}\right)\right)\right) \]

       5. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(C \cdot F\right), \frac{1}{2}\right), \left(\mathsf{neg}\left(\color{blue}{\frac{2}{B}}\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(F \cdot C\right), \frac{1}{2}\right), \left(\mathsf{neg}\left(\frac{\color{blue}{2}}{B}\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(F, C\right), \frac{1}{2}\right), \left(\mathsf{neg}\left(\frac{\color{blue}{2}}{B}\right)\right)\right) \]

       8. distribute-neg-frac N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(F, C\right), \frac{1}{2}\right), \left(\frac{\mathsf{neg}\left(2\right)}{\color{blue}{B}}\right)\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(F, C\right), \frac{1}{2}\right), \mathsf{/.f64}\left(\left(\mathsf{neg}\left(2\right)\right), \color{blue}{B}\right)\right) \]

       10. metadata-eval 7.5%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(F, C\right), \frac{1}{2}\right), \mathsf{/.f64}\left(-2, B\right)\right) \]

   12. Applied egg-rr 7.5%

       \[\leadsto \color{blue}{{\left(F \cdot C\right)}^{0.5} \cdot \frac{-2}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 5.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -1.3 \cdot 10^{-304}:\\ \;\;\;\;0.25 \cdot \left(\sqrt{B \cdot F} \cdot \frac{1}{C}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(C \cdot F\right)}^{0.5} \cdot \frac{-2}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 5.0% accurate, 5.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \sqrt{C \cdot F} \cdot \left(0 - \frac{2}{B\_m}\right) \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F) :precision binary64 (* (sqrt (* C F)) (- 0.0 (/ 2.0 B_m))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	return sqrt((C * F)) * (0.0 - (2.0 / 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 = sqrt((c * f)) * (0.0d0 - (2.0d0 / 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.sqrt((C * F)) * (0.0 - (2.0 / B_m));
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	return math.sqrt((C * F)) * (0.0 - (2.0 / B_m))

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	return Float64(sqrt(Float64(C * F)) * Float64(0.0 - Float64(2.0 / B_m)))
end

MATLAB

B_m = abs(B);
function tmp = code(A, B_m, C, F)
	tmp = sqrt((C * F)) * (0.0 - (2.0 / B_m));
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := N[(N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision] * N[(0.0 - N[(2.0 / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 22.2%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Add Preprocessing

3. Taylor expanded in A around 0

   \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
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. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(C, \left(\sqrt{{C}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(C, \left(\sqrt{C \cdot C + {B}^{2}}\right)\right)\right)\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(C, \left(\sqrt{C \cdot C + B \cdot B}\right)\right)\right)\right)\right) \]

   12. hypot-define N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(C, \left(\mathsf{hypot}\left(C, B\right)\right)\right)\right)\right)\right) \]

   13. hypot-lowering-hypot.f64 16.0%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(C, B\right)\right)\right)\right)\right) \]

5. Simplified 16.0%

   \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)}} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \sqrt{F \cdot \left(C + \sqrt{C \cdot C + B \cdot B}\right)} \cdot \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right)} \]

   2. pow1/2 N/A

      \[\leadsto {\left(F \cdot \left(C + \sqrt{C \cdot C + B \cdot B}\right)\right)}^{\frac{1}{2}} \cdot \left(\color{blue}{-1} \cdot \frac{\sqrt{2}}{B}\right) \]

   3. unpow-prod-down N/A

      \[\leadsto \left({F}^{\frac{1}{2}} \cdot {\left(C + \sqrt{C \cdot C + B \cdot B}\right)}^{\frac{1}{2}}\right) \cdot \left(\color{blue}{-1} \cdot \frac{\sqrt{2}}{B}\right) \]

   4. associate-*l* N/A

      \[\leadsto {F}^{\frac{1}{2}} \cdot \color{blue}{\left({\left(C + \sqrt{C \cdot C + B \cdot B}\right)}^{\frac{1}{2}} \cdot \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right)} \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left({F}^{\frac{1}{2}}\right), \color{blue}{\left({\left(C + \sqrt{C \cdot C + B \cdot B}\right)}^{\frac{1}{2}} \cdot \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right)}\right) \]

   6. pow1/2 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{F}\right), \left(\color{blue}{{\left(C + \sqrt{C \cdot C + B \cdot B}\right)}^{\frac{1}{2}}} \cdot \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right)\right) \]

   7. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \left(\color{blue}{{\left(C + \sqrt{C \cdot C + B \cdot B}\right)}^{\frac{1}{2}}} \cdot \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\left({\left(C + \sqrt{C \cdot C + B \cdot B}\right)}^{\frac{1}{2}}\right), \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right)}\right)\right) \]

   9. pow1/2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\left(\sqrt{C + \sqrt{C \cdot C + B \cdot B}}\right), \left(\color{blue}{-1} \cdot \frac{\sqrt{2}}{B}\right)\right)\right) \]

   10. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(C + \sqrt{C \cdot C + B \cdot B}\right)\right), \left(\color{blue}{-1} \cdot \frac{\sqrt{2}}{B}\right)\right)\right) \]

   11. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \left(\sqrt{C \cdot C + B \cdot B}\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right)\right) \]

   12. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \left(\sqrt{B \cdot B + C \cdot C}\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right)\right) \]

   13. hypot-define N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \left(\mathsf{hypot}\left(B, C\right)\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right)\right) \]

   14. hypot-lowering-hypot.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right)\right) \]

   15. mul-1-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right), \left(\mathsf{neg}\left(\frac{\sqrt{2}}{B}\right)\right)\right)\right) \]

   16. neg-sub0 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right), \left(0 - \color{blue}{\frac{\sqrt{2}}{B}}\right)\right)\right) \]

   17. --lowering--.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right), \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{\sqrt{2}}{B}\right)}\right)\right)\right) \]

   18. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left(\sqrt{2}\right), \color{blue}{B}\right)\right)\right)\right) \]

7. Applied egg-rr 21.6%

   \[\leadsto \color{blue}{\sqrt{F} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \left(0 - \frac{\sqrt{2}}{B}\right)\right)} \]

8. Taylor expanded in C around inf

   \[\leadsto \color{blue}{-1 \cdot \left(\frac{{\left(\sqrt{2}\right)}^{2}}{B} \cdot \sqrt{C \cdot F}\right)} \]
9. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{{\left(\sqrt{2}\right)}^{2}}{B} \cdot \sqrt{C \cdot F}\right) \]

   2. neg-lowering-neg.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\left(\frac{{\left(\sqrt{2}\right)}^{2}}{B} \cdot \sqrt{C \cdot F}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\frac{{\left(\sqrt{2}\right)}^{2}}{B}\right), \left(\sqrt{C \cdot F}\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\frac{\sqrt{2} \cdot \sqrt{2}}{B}\right), \left(\sqrt{C \cdot F}\right)\right)\right) \]

   5. rem-square-sqrt N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\frac{2}{B}\right), \left(\sqrt{C \cdot F}\right)\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, B\right), \left(\sqrt{C \cdot F}\right)\right)\right) \]

   7. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, B\right), \mathsf{sqrt.f64}\left(\left(C \cdot F\right)\right)\right)\right) \]

   8. *-lowering-*.f64 4.3%

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(C, F\right)\right)\right)\right) \]

10. Simplified 4.3%

    \[\leadsto \color{blue}{-\frac{2}{B} \cdot \sqrt{C \cdot F}} \]

11. Final simplification 4.3%

    \[\leadsto \sqrt{C \cdot F} \cdot \left(0 - \frac{2}{B}\right) \]
12. Add Preprocessing

Alternative 26: 5.2% accurate, 5.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ {\left(C \cdot F\right)}^{0.5} \cdot \frac{-2}{B\_m} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F) :precision binary64 (* (pow (* C F) 0.5) (/ -2.0 B_m)))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	return pow((C * F), 0.5) * (-2.0 / 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 = ((c * f) ** 0.5d0) * ((-2.0d0) / 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((C * F), 0.5) * (-2.0 / B_m);
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	return math.pow((C * F), 0.5) * (-2.0 / B_m)

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	return Float64((Float64(C * F) ^ 0.5) * Float64(-2.0 / B_m))
end

MATLAB

B_m = abs(B);
function tmp = code(A, B_m, C, F)
	tmp = ((C * F) ^ 0.5) * (-2.0 / B_m);
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := N[(N[Power[N[(C * F), $MachinePrecision], 0.5], $MachinePrecision] * N[(-2.0 / B$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 22.2%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Add Preprocessing

3. Taylor expanded in A around 0

   \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
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. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(C, \left(\sqrt{{C}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(C, \left(\sqrt{C \cdot C + {B}^{2}}\right)\right)\right)\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(C, \left(\sqrt{C \cdot C + B \cdot B}\right)\right)\right)\right)\right) \]

   12. hypot-define N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(C, \left(\mathsf{hypot}\left(C, B\right)\right)\right)\right)\right)\right) \]

   13. hypot-lowering-hypot.f64 16.0%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(C, B\right)\right)\right)\right)\right) \]

5. Simplified 16.0%

   \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)}} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \sqrt{F \cdot \left(C + \sqrt{C \cdot C + B \cdot B}\right)} \cdot \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right)} \]

   2. pow1/2 N/A

      \[\leadsto {\left(F \cdot \left(C + \sqrt{C \cdot C + B \cdot B}\right)\right)}^{\frac{1}{2}} \cdot \left(\color{blue}{-1} \cdot \frac{\sqrt{2}}{B}\right) \]

   3. unpow-prod-down N/A

      \[\leadsto \left({F}^{\frac{1}{2}} \cdot {\left(C + \sqrt{C \cdot C + B \cdot B}\right)}^{\frac{1}{2}}\right) \cdot \left(\color{blue}{-1} \cdot \frac{\sqrt{2}}{B}\right) \]

   4. associate-*l* N/A

      \[\leadsto {F}^{\frac{1}{2}} \cdot \color{blue}{\left({\left(C + \sqrt{C \cdot C + B \cdot B}\right)}^{\frac{1}{2}} \cdot \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right)} \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left({F}^{\frac{1}{2}}\right), \color{blue}{\left({\left(C + \sqrt{C \cdot C + B \cdot B}\right)}^{\frac{1}{2}} \cdot \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right)}\right) \]

   6. pow1/2 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{F}\right), \left(\color{blue}{{\left(C + \sqrt{C \cdot C + B \cdot B}\right)}^{\frac{1}{2}}} \cdot \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right)\right) \]

   7. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \left(\color{blue}{{\left(C + \sqrt{C \cdot C + B \cdot B}\right)}^{\frac{1}{2}}} \cdot \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\left({\left(C + \sqrt{C \cdot C + B \cdot B}\right)}^{\frac{1}{2}}\right), \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right)}\right)\right) \]

   9. pow1/2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\left(\sqrt{C + \sqrt{C \cdot C + B \cdot B}}\right), \left(\color{blue}{-1} \cdot \frac{\sqrt{2}}{B}\right)\right)\right) \]

   10. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(C + \sqrt{C \cdot C + B \cdot B}\right)\right), \left(\color{blue}{-1} \cdot \frac{\sqrt{2}}{B}\right)\right)\right) \]

   11. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \left(\sqrt{C \cdot C + B \cdot B}\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right)\right) \]

   12. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \left(\sqrt{B \cdot B + C \cdot C}\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right)\right) \]

   13. hypot-define N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \left(\mathsf{hypot}\left(B, C\right)\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right)\right) \]

   14. hypot-lowering-hypot.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right)\right) \]

   15. mul-1-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right), \left(\mathsf{neg}\left(\frac{\sqrt{2}}{B}\right)\right)\right)\right) \]

   16. neg-sub0 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right), \left(0 - \color{blue}{\frac{\sqrt{2}}{B}}\right)\right)\right) \]

   17. --lowering--.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right), \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{\sqrt{2}}{B}\right)}\right)\right)\right) \]

   18. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left(\sqrt{2}\right), \color{blue}{B}\right)\right)\right)\right) \]

7. Applied egg-rr 21.6%

   \[\leadsto \color{blue}{\sqrt{F} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \left(0 - \frac{\sqrt{2}}{B}\right)\right)} \]

8. Taylor expanded in C around inf

   \[\leadsto \color{blue}{-1 \cdot \left(\frac{{\left(\sqrt{2}\right)}^{2}}{B} \cdot \sqrt{C \cdot F}\right)} \]
9. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{{\left(\sqrt{2}\right)}^{2}}{B} \cdot \sqrt{C \cdot F}\right) \]

   2. neg-lowering-neg.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\left(\frac{{\left(\sqrt{2}\right)}^{2}}{B} \cdot \sqrt{C \cdot F}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\frac{{\left(\sqrt{2}\right)}^{2}}{B}\right), \left(\sqrt{C \cdot F}\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\frac{\sqrt{2} \cdot \sqrt{2}}{B}\right), \left(\sqrt{C \cdot F}\right)\right)\right) \]

   5. rem-square-sqrt N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\frac{2}{B}\right), \left(\sqrt{C \cdot F}\right)\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, B\right), \left(\sqrt{C \cdot F}\right)\right)\right) \]

   7. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, B\right), \mathsf{sqrt.f64}\left(\left(C \cdot F\right)\right)\right)\right) \]

   8. *-lowering-*.f64 4.3%

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(C, F\right)\right)\right)\right) \]

10. Simplified 4.3%

    \[\leadsto \color{blue}{-\frac{2}{B} \cdot \sqrt{C \cdot F}} \]
11. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{C \cdot F} \cdot \frac{2}{B}\right) \]

    2. distribute-rgt-neg-in N/A

       \[\leadsto \sqrt{C \cdot F} \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{2}{B}\right)\right)} \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{C \cdot F}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{2}{B}\right)\right)}\right) \]

    4. pow1/2 N/A

       \[\leadsto \mathsf{*.f64}\left(\left({\left(C \cdot F\right)}^{\frac{1}{2}}\right), \left(\mathsf{neg}\left(\color{blue}{\frac{2}{B}}\right)\right)\right) \]

    5. pow-lowering-pow.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(C \cdot F\right), \frac{1}{2}\right), \left(\mathsf{neg}\left(\color{blue}{\frac{2}{B}}\right)\right)\right) \]

    6. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(F \cdot C\right), \frac{1}{2}\right), \left(\mathsf{neg}\left(\frac{\color{blue}{2}}{B}\right)\right)\right) \]

    7. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(F, C\right), \frac{1}{2}\right), \left(\mathsf{neg}\left(\frac{\color{blue}{2}}{B}\right)\right)\right) \]

    8. distribute-neg-frac N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(F, C\right), \frac{1}{2}\right), \left(\frac{\mathsf{neg}\left(2\right)}{\color{blue}{B}}\right)\right) \]

    9. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(F, C\right), \frac{1}{2}\right), \mathsf{/.f64}\left(\left(\mathsf{neg}\left(2\right)\right), \color{blue}{B}\right)\right) \]

    10. metadata-eval 4.5%

        \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(F, C\right), \frac{1}{2}\right), \mathsf{/.f64}\left(-2, B\right)\right) \]

12. Applied egg-rr 4.5%

    \[\leadsto \color{blue}{{\left(F \cdot C\right)}^{0.5} \cdot \frac{-2}{B}} \]

13. Final simplification 4.5%

    \[\leadsto {\left(C \cdot F\right)}^{0.5} \cdot \frac{-2}{B} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024152 
(FPCore (A B C F)
  :name "ABCF->ab-angle a"
  :precision binary64
  (/ (- (sqrt (* (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F)) (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))) (- (pow B 2.0) (* (* 4.0 A) C))))