ABCF->ab-angle a

Percentage Accurate: 19.2% → 51.3%

Time: 29.9s

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: 19.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 51.3% accurate, 0.4× speedup

Math

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

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (hypot B_m (- A C)))
        (t_1 (* (* 4.0 A) C))
        (t_2
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_1) F))
            (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_1 (pow B_m 2.0)))))
   (if (<= t_2 0.0)
     (/
      (* (sqrt (* 2.0 (+ (* B_m B_m) (* -4.0 (* A C))))) (sqrt F))
      (* (pow (+ C (+ A t_0)) -0.5) (- (* 4.0 (* A C)) (* B_m B_m))))
     (if (<= t_2 INFINITY)
       (/
        (*
         (* (pow (* F -4.0) 0.5) (sqrt (* A C)))
         (sqrt (* 2.0 (+ A (+ C t_0)))))
        (- t_1 (* 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 t_0 = hypot(B_m, (A - C));
	double t_1 = (4.0 * A) * C;
	double t_2 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_1) * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_1 - pow(B_m, 2.0));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = (sqrt((2.0 * ((B_m * B_m) + (-4.0 * (A * C))))) * sqrt(F)) / (pow((C + (A + t_0)), -0.5) * ((4.0 * (A * C)) - (B_m * B_m)));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = ((pow((F * -4.0), 0.5) * sqrt((A * C))) * sqrt((2.0 * (A + (C + t_0))))) / (t_1 - (B_m * B_m));
	} 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 t_0 = Math.hypot(B_m, (A - C));
	double t_1 = (4.0 * A) * C;
	double t_2 = Math.sqrt(((2.0 * ((Math.pow(B_m, 2.0) - t_1) * F)) * ((A + C) + Math.sqrt((Math.pow(B_m, 2.0) + Math.pow((A - C), 2.0)))))) / (t_1 - Math.pow(B_m, 2.0));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = (Math.sqrt((2.0 * ((B_m * B_m) + (-4.0 * (A * C))))) * Math.sqrt(F)) / (Math.pow((C + (A + t_0)), -0.5) * ((4.0 * (A * C)) - (B_m * B_m)));
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = ((Math.pow((F * -4.0), 0.5) * Math.sqrt((A * C))) * Math.sqrt((2.0 * (A + (C + t_0))))) / (t_1 - (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):
	t_0 = math.hypot(B_m, (A - C))
	t_1 = (4.0 * A) * C
	t_2 = math.sqrt(((2.0 * ((math.pow(B_m, 2.0) - t_1) * F)) * ((A + C) + math.sqrt((math.pow(B_m, 2.0) + math.pow((A - C), 2.0)))))) / (t_1 - math.pow(B_m, 2.0))
	tmp = 0
	if t_2 <= 0.0:
		tmp = (math.sqrt((2.0 * ((B_m * B_m) + (-4.0 * (A * C))))) * math.sqrt(F)) / (math.pow((C + (A + t_0)), -0.5) * ((4.0 * (A * C)) - (B_m * B_m)))
	elif t_2 <= math.inf:
		tmp = ((math.pow((F * -4.0), 0.5) * math.sqrt((A * C))) * math.sqrt((2.0 * (A + (C + t_0))))) / (t_1 - (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)
	t_0 = hypot(B_m, Float64(A - C))
	t_1 = Float64(Float64(4.0 * A) * C)
	t_2 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_1) * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_1 - (B_m ^ 2.0)))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(Float64(B_m * B_m) + Float64(-4.0 * Float64(A * C))))) * sqrt(F)) / Float64((Float64(C + Float64(A + t_0)) ^ -0.5) * Float64(Float64(4.0 * Float64(A * C)) - Float64(B_m * B_m))));
	elseif (t_2 <= Inf)
		tmp = Float64(Float64(Float64((Float64(F * -4.0) ^ 0.5) * sqrt(Float64(A * C))) * sqrt(Float64(2.0 * Float64(A + Float64(C + t_0))))) / Float64(t_1 - 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)
	t_0 = hypot(B_m, (A - C));
	t_1 = (4.0 * A) * C;
	t_2 = sqrt(((2.0 * (((B_m ^ 2.0) - t_1) * F)) * ((A + C) + sqrt(((B_m ^ 2.0) + ((A - C) ^ 2.0)))))) / (t_1 - (B_m ^ 2.0));
	tmp = 0.0;
	if (t_2 <= 0.0)
		tmp = (sqrt((2.0 * ((B_m * B_m) + (-4.0 * (A * C))))) * sqrt(F)) / (((C + (A + t_0)) ^ -0.5) * ((4.0 * (A * C)) - (B_m * B_m)));
	elseif (t_2 <= Inf)
		tmp = ((((F * -4.0) ^ 0.5) * sqrt((A * C))) * sqrt((2.0 * (A + (C + t_0))))) / (t_1 - (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_] := Block[{t$95$0 = N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$1), $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$1 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(N[(N[Sqrt[N[(2.0 * N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[(C + N[(A + t$95$0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[(N[(N[Power[N[(F * -4.0), $MachinePrecision], 0.5], $MachinePrecision] * N[Sqrt[N[(A * C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(2.0 * N[(A + N[(C + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(t$95$1 - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision] * N[(0.0 - N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -0.0

   1. Initial program 29.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 37.7%

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

      1. pow1/2 N/A

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

      2. unpow-prod-down N/A

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

      3. associate-/l* N/A

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

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

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

   6. Applied egg-rr 50.4%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

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

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

   8. Applied egg-rr 50.4%

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

      1. pow1/2 N/A

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

      2. associate-*r* N/A

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

      3. unpow-prod-down N/A

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

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

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

      5. pow1/2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. pow1/2 N/A

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

      13. sqrt-lowering-sqrt.f64 62.9%

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

   10. Applied egg-rr 62.9%

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

       1. clear-num N/A

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

       2. associate-/r/ N/A

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

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

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

   12. Applied egg-rr 63.2%

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

   if -0.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 33.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. 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 53.9%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   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 74.7%

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

   7. Taylor expanded in B around 0

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

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

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

      2. *-lowering-*.f64 74.7%

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

   9. Simplified 74.7%

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

       1. pow1/2 N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. unpow-prod-down N/A

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

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

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

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

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

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

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

       8. pow1/2 N/A

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

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

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

       10. *-lowering-*.f64 82.6%

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

   11. Applied egg-rr 82.6%

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

   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 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 15.6%

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

   5. Simplified 15.6%

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

4. Final simplification 45.0%

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

Alternative 2: 46.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \frac{A \cdot \left(4 \cdot C\right) - B\_m \cdot B\_m}{\sqrt{A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)}}\\ t_1 := B\_m \cdot B\_m + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B\_m \leq 8.5 \cdot 10^{-100}:\\ \;\;\;\;\frac{\sqrt{t\_1 \cdot \left(2 \cdot F\right)}}{t\_0}\\ \mathbf{elif}\;B\_m \leq 1.05 \cdot 10^{+153}:\\ \;\;\;\;\frac{\sqrt{2 \cdot t\_1} \cdot \sqrt{F}}{t\_0}\\ \mathbf{elif}\;B\_m \leq 2.2 \cdot 10^{+224}:\\ \;\;\;\;\left(0 - \frac{\sqrt{2}}{B\_m}\right) \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B\_m}} \cdot \left(0 - \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0
         (/
          (- (* A (* 4.0 C)) (* B_m B_m))
          (sqrt (+ A (+ C (hypot B_m (- A C)))))))
        (t_1 (+ (* B_m B_m) (* -4.0 (* A C)))))
   (if (<= B_m 8.5e-100)
     (/ (sqrt (* t_1 (* 2.0 F))) t_0)
     (if (<= B_m 1.05e+153)
       (/ (* (sqrt (* 2.0 t_1)) (sqrt F)) t_0)
       (if (<= B_m 2.2e+224)
         (* (- 0.0 (/ (sqrt 2.0) B_m)) (sqrt (* F (+ C (hypot C 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 t_0 = ((A * (4.0 * C)) - (B_m * B_m)) / sqrt((A + (C + hypot(B_m, (A - C)))));
	double t_1 = (B_m * B_m) + (-4.0 * (A * C));
	double tmp;
	if (B_m <= 8.5e-100) {
		tmp = sqrt((t_1 * (2.0 * F))) / t_0;
	} else if (B_m <= 1.05e+153) {
		tmp = (sqrt((2.0 * t_1)) * sqrt(F)) / t_0;
	} else if (B_m <= 2.2e+224) {
		tmp = (0.0 - (sqrt(2.0) / B_m)) * sqrt((F * (C + hypot(C, B_m))));
	} 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 t_0 = ((A * (4.0 * C)) - (B_m * B_m)) / Math.sqrt((A + (C + Math.hypot(B_m, (A - C)))));
	double t_1 = (B_m * B_m) + (-4.0 * (A * C));
	double tmp;
	if (B_m <= 8.5e-100) {
		tmp = Math.sqrt((t_1 * (2.0 * F))) / t_0;
	} else if (B_m <= 1.05e+153) {
		tmp = (Math.sqrt((2.0 * t_1)) * Math.sqrt(F)) / t_0;
	} else if (B_m <= 2.2e+224) {
		tmp = (0.0 - (Math.sqrt(2.0) / B_m)) * Math.sqrt((F * (C + Math.hypot(C, 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):
	t_0 = ((A * (4.0 * C)) - (B_m * B_m)) / math.sqrt((A + (C + math.hypot(B_m, (A - C)))))
	t_1 = (B_m * B_m) + (-4.0 * (A * C))
	tmp = 0
	if B_m <= 8.5e-100:
		tmp = math.sqrt((t_1 * (2.0 * F))) / t_0
	elif B_m <= 1.05e+153:
		tmp = (math.sqrt((2.0 * t_1)) * math.sqrt(F)) / t_0
	elif B_m <= 2.2e+224:
		tmp = (0.0 - (math.sqrt(2.0) / B_m)) * math.sqrt((F * (C + math.hypot(C, 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)
	t_0 = Float64(Float64(Float64(A * Float64(4.0 * C)) - Float64(B_m * B_m)) / sqrt(Float64(A + Float64(C + 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 <= 8.5e-100)
		tmp = Float64(sqrt(Float64(t_1 * Float64(2.0 * F))) / t_0);
	elseif (B_m <= 1.05e+153)
		tmp = Float64(Float64(sqrt(Float64(2.0 * t_1)) * sqrt(F)) / t_0);
	elseif (B_m <= 2.2e+224)
		tmp = Float64(Float64(0.0 - Float64(sqrt(2.0) / B_m)) * sqrt(Float64(F * Float64(C + hypot(C, 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)
	t_0 = ((A * (4.0 * C)) - (B_m * B_m)) / sqrt((A + (C + hypot(B_m, (A - C)))));
	t_1 = (B_m * B_m) + (-4.0 * (A * C));
	tmp = 0.0;
	if (B_m <= 8.5e-100)
		tmp = sqrt((t_1 * (2.0 * F))) / t_0;
	elseif (B_m <= 1.05e+153)
		tmp = (sqrt((2.0 * t_1)) * sqrt(F)) / t_0;
	elseif (B_m <= 2.2e+224)
		tmp = (0.0 - (sqrt(2.0) / B_m)) * sqrt((F * (C + hypot(C, 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_] := Block[{t$95$0 = N[(N[(N[(A * N[(4.0 * C), $MachinePrecision]), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $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, 8.5e-100], N[(N[Sqrt[N[(t$95$1 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[B$95$m, 1.05e+153], N[(N[(N[Sqrt[N[(2.0 * t$95$1), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[B$95$m, 2.2e+224], N[(N[(0.0 - N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(F * N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision] * N[(0.0 - N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 8.50000000000000017e-100

   1. Initial program 18.3%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 26.3%

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

      1. pow1/2 N/A

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

      2. unpow-prod-down N/A

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

      3. associate-/l* N/A

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

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

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

   6. Applied egg-rr 33.8%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

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

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

   8. Applied egg-rr 33.9%

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

   if 8.50000000000000017e-100 < B < 1.05000000000000008e153

   1. Initial program 22.5%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 26.4%

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

      1. pow1/2 N/A

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

      2. unpow-prod-down N/A

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

      3. associate-/l* N/A

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

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

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

   6. Applied egg-rr 40.6%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

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

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

   8. Applied egg-rr 40.6%

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

      1. pow1/2 N/A

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

      2. associate-*r* N/A

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

      3. unpow-prod-down N/A

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

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

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

      5. pow1/2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. pow1/2 N/A

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

      13. sqrt-lowering-sqrt.f64 49.5%

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

   10. Applied egg-rr 49.5%

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

   if 1.05000000000000008e153 < B < 2.2e224

   1. Initial program 8.9%

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

   3. Taylor expanded in A around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. +-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 55.7%

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

   5. Simplified 55.7%

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

   if 2.2e224 < B

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

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

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

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

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

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

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

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

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

      6. sqrt-lowering-sqrt.f64 67.0%

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

   5. Simplified 67.0%

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

4. Final simplification 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 8.5 \cdot 10^{-100}:\\ \;\;\;\;\frac{\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)}}{\frac{A \cdot \left(4 \cdot C\right) - B \cdot B}{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}}}\\ \mathbf{elif}\;B \leq 1.05 \cdot 10^{+153}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)} \cdot \sqrt{F}}{\frac{A \cdot \left(4 \cdot C\right) - B \cdot B}{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}}}\\ \mathbf{elif}\;B \leq 2.2 \cdot 10^{+224}:\\ \;\;\;\;\left(0 - \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(0 - \sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 45.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := A \cdot \left(4 \cdot C\right) - B\_m \cdot B\_m\\ t_1 := B\_m \cdot B\_m + -4 \cdot \left(A \cdot C\right)\\ t_2 := A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)\\ \mathbf{if}\;B\_m \leq 2.1 \cdot 10^{-99}:\\ \;\;\;\;\frac{\sqrt{t\_1 \cdot \left(2 \cdot F\right)}}{\frac{t\_0}{\sqrt{t\_2}}}\\ \mathbf{elif}\;B\_m \leq 4.5 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{t\_1} \cdot \left(\sqrt{F} \cdot \frac{\sqrt{2 \cdot t\_2}}{t\_0}\right)\\ \mathbf{elif}\;B\_m \leq 1.12 \cdot 10^{+223}:\\ \;\;\;\;\left(0 - \frac{\sqrt{2}}{B\_m}\right) \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B\_m}} \cdot \left(0 - \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (* A (* 4.0 C)) (* B_m B_m)))
        (t_1 (+ (* B_m B_m) (* -4.0 (* A C))))
        (t_2 (+ A (+ C (hypot B_m (- A C))))))
   (if (<= B_m 2.1e-99)
     (/ (sqrt (* t_1 (* 2.0 F))) (/ t_0 (sqrt t_2)))
     (if (<= B_m 4.5e+152)
       (* (sqrt t_1) (* (sqrt F) (/ (sqrt (* 2.0 t_2)) t_0)))
       (if (<= B_m 1.12e+223)
         (* (- 0.0 (/ (sqrt 2.0) B_m)) (sqrt (* F (+ C (hypot C 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 t_0 = (A * (4.0 * C)) - (B_m * B_m);
	double t_1 = (B_m * B_m) + (-4.0 * (A * C));
	double t_2 = A + (C + hypot(B_m, (A - C)));
	double tmp;
	if (B_m <= 2.1e-99) {
		tmp = sqrt((t_1 * (2.0 * F))) / (t_0 / sqrt(t_2));
	} else if (B_m <= 4.5e+152) {
		tmp = sqrt(t_1) * (sqrt(F) * (sqrt((2.0 * t_2)) / t_0));
	} else if (B_m <= 1.12e+223) {
		tmp = (0.0 - (sqrt(2.0) / B_m)) * sqrt((F * (C + hypot(C, B_m))));
	} 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 t_0 = (A * (4.0 * C)) - (B_m * B_m);
	double t_1 = (B_m * B_m) + (-4.0 * (A * C));
	double t_2 = A + (C + Math.hypot(B_m, (A - C)));
	double tmp;
	if (B_m <= 2.1e-99) {
		tmp = Math.sqrt((t_1 * (2.0 * F))) / (t_0 / Math.sqrt(t_2));
	} else if (B_m <= 4.5e+152) {
		tmp = Math.sqrt(t_1) * (Math.sqrt(F) * (Math.sqrt((2.0 * t_2)) / t_0));
	} else if (B_m <= 1.12e+223) {
		tmp = (0.0 - (Math.sqrt(2.0) / B_m)) * Math.sqrt((F * (C + Math.hypot(C, 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):
	t_0 = (A * (4.0 * C)) - (B_m * B_m)
	t_1 = (B_m * B_m) + (-4.0 * (A * C))
	t_2 = A + (C + math.hypot(B_m, (A - C)))
	tmp = 0
	if B_m <= 2.1e-99:
		tmp = math.sqrt((t_1 * (2.0 * F))) / (t_0 / math.sqrt(t_2))
	elif B_m <= 4.5e+152:
		tmp = math.sqrt(t_1) * (math.sqrt(F) * (math.sqrt((2.0 * t_2)) / t_0))
	elif B_m <= 1.12e+223:
		tmp = (0.0 - (math.sqrt(2.0) / B_m)) * math.sqrt((F * (C + math.hypot(C, 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)
	t_0 = Float64(Float64(A * Float64(4.0 * C)) - Float64(B_m * B_m))
	t_1 = Float64(Float64(B_m * B_m) + Float64(-4.0 * Float64(A * C)))
	t_2 = Float64(A + Float64(C + hypot(B_m, Float64(A - C))))
	tmp = 0.0
	if (B_m <= 2.1e-99)
		tmp = Float64(sqrt(Float64(t_1 * Float64(2.0 * F))) / Float64(t_0 / sqrt(t_2)));
	elseif (B_m <= 4.5e+152)
		tmp = Float64(sqrt(t_1) * Float64(sqrt(F) * Float64(sqrt(Float64(2.0 * t_2)) / t_0)));
	elseif (B_m <= 1.12e+223)
		tmp = Float64(Float64(0.0 - Float64(sqrt(2.0) / B_m)) * sqrt(Float64(F * Float64(C + hypot(C, 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)
	t_0 = (A * (4.0 * C)) - (B_m * B_m);
	t_1 = (B_m * B_m) + (-4.0 * (A * C));
	t_2 = A + (C + hypot(B_m, (A - C)));
	tmp = 0.0;
	if (B_m <= 2.1e-99)
		tmp = sqrt((t_1 * (2.0 * F))) / (t_0 / sqrt(t_2));
	elseif (B_m <= 4.5e+152)
		tmp = sqrt(t_1) * (sqrt(F) * (sqrt((2.0 * t_2)) / t_0));
	elseif (B_m <= 1.12e+223)
		tmp = (0.0 - (sqrt(2.0) / B_m)) * sqrt((F * (C + hypot(C, 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_] := Block[{t$95$0 = N[(N[(A * N[(4.0 * C), $MachinePrecision]), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 2.1e-99], N[(N[Sqrt[N[(t$95$1 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 / N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 4.5e+152], N[(N[Sqrt[t$95$1], $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * N[(N[Sqrt[N[(2.0 * t$95$2), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 1.12e+223], N[(N[(0.0 - N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(F * N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision] * N[(0.0 - N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 2.09999999999999984e-99

   1. Initial program 18.3%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 26.3%

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

      1. pow1/2 N/A

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

      2. unpow-prod-down N/A

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

      3. associate-/l* N/A

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

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

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

   6. Applied egg-rr 33.8%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

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

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

   8. Applied egg-rr 33.9%

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

   if 2.09999999999999984e-99 < B < 4.5000000000000001e152

   1. Initial program 22.5%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 26.4%

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

      1. pow1/2 N/A

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

      2. unpow-prod-down N/A

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

      3. associate-/l* N/A

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

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

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

   6. Applied egg-rr 40.6%

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

   8. Applied egg-rr 49.6%

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

   if 4.5000000000000001e152 < B < 1.1200000000000001e223

   1. Initial program 8.9%

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

   3. Taylor expanded in A around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. +-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 55.7%

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

   5. Simplified 55.7%

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

   if 1.1200000000000001e223 < B

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

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

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

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

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

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

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

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

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

      6. sqrt-lowering-sqrt.f64 67.0%

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

   5. Simplified 67.0%

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

4. Final simplification 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.1 \cdot 10^{-99}:\\ \;\;\;\;\frac{\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)}}{\frac{A \cdot \left(4 \cdot C\right) - B \cdot B}{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}}}\\ \mathbf{elif}\;B \leq 4.5 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{B \cdot B + -4 \cdot \left(A \cdot C\right)} \cdot \left(\sqrt{F} \cdot \frac{\sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{A \cdot \left(4 \cdot C\right) - B \cdot B}\right)\\ \mathbf{elif}\;B \leq 1.12 \cdot 10^{+223}:\\ \;\;\;\;\left(0 - \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(0 - \sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 46.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \frac{A \cdot \left(4 \cdot C\right) - B\_m \cdot B\_m}{\sqrt{A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)}}\\ \mathbf{if}\;B\_m \leq 3.6 \cdot 10^{+43}:\\ \;\;\;\;\frac{\sqrt{\left(B\_m \cdot B\_m + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)}}{t\_0}\\ \mathbf{elif}\;B\_m \leq 1.05 \cdot 10^{+153}:\\ \;\;\;\;\frac{\sqrt{F} \cdot \left(B\_m \cdot \sqrt{2}\right)}{t\_0}\\ \mathbf{elif}\;B\_m \leq 1.2 \cdot 10^{+224}:\\ \;\;\;\;\left(0 - \frac{\sqrt{2}}{B\_m}\right) \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B\_m}} \cdot \left(0 - \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0
         (/
          (- (* A (* 4.0 C)) (* B_m B_m))
          (sqrt (+ A (+ C (hypot B_m (- A C))))))))
   (if (<= B_m 3.6e+43)
     (/ (sqrt (* (+ (* B_m B_m) (* -4.0 (* A C))) (* 2.0 F))) t_0)
     (if (<= B_m 1.05e+153)
       (/ (* (sqrt F) (* B_m (sqrt 2.0))) t_0)
       (if (<= B_m 1.2e+224)
         (* (- 0.0 (/ (sqrt 2.0) B_m)) (sqrt (* F (+ C (hypot C 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 t_0 = ((A * (4.0 * C)) - (B_m * B_m)) / sqrt((A + (C + hypot(B_m, (A - C)))));
	double tmp;
	if (B_m <= 3.6e+43) {
		tmp = sqrt((((B_m * B_m) + (-4.0 * (A * C))) * (2.0 * F))) / t_0;
	} else if (B_m <= 1.05e+153) {
		tmp = (sqrt(F) * (B_m * sqrt(2.0))) / t_0;
	} else if (B_m <= 1.2e+224) {
		tmp = (0.0 - (sqrt(2.0) / B_m)) * sqrt((F * (C + hypot(C, B_m))));
	} 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 t_0 = ((A * (4.0 * C)) - (B_m * B_m)) / Math.sqrt((A + (C + Math.hypot(B_m, (A - C)))));
	double tmp;
	if (B_m <= 3.6e+43) {
		tmp = Math.sqrt((((B_m * B_m) + (-4.0 * (A * C))) * (2.0 * F))) / t_0;
	} else if (B_m <= 1.05e+153) {
		tmp = (Math.sqrt(F) * (B_m * Math.sqrt(2.0))) / t_0;
	} else if (B_m <= 1.2e+224) {
		tmp = (0.0 - (Math.sqrt(2.0) / B_m)) * Math.sqrt((F * (C + Math.hypot(C, 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):
	t_0 = ((A * (4.0 * C)) - (B_m * B_m)) / math.sqrt((A + (C + math.hypot(B_m, (A - C)))))
	tmp = 0
	if B_m <= 3.6e+43:
		tmp = math.sqrt((((B_m * B_m) + (-4.0 * (A * C))) * (2.0 * F))) / t_0
	elif B_m <= 1.05e+153:
		tmp = (math.sqrt(F) * (B_m * math.sqrt(2.0))) / t_0
	elif B_m <= 1.2e+224:
		tmp = (0.0 - (math.sqrt(2.0) / B_m)) * math.sqrt((F * (C + math.hypot(C, 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)
	t_0 = Float64(Float64(Float64(A * Float64(4.0 * C)) - Float64(B_m * B_m)) / sqrt(Float64(A + Float64(C + hypot(B_m, Float64(A - C))))))
	tmp = 0.0
	if (B_m <= 3.6e+43)
		tmp = Float64(sqrt(Float64(Float64(Float64(B_m * B_m) + Float64(-4.0 * Float64(A * C))) * Float64(2.0 * F))) / t_0);
	elseif (B_m <= 1.05e+153)
		tmp = Float64(Float64(sqrt(F) * Float64(B_m * sqrt(2.0))) / t_0);
	elseif (B_m <= 1.2e+224)
		tmp = Float64(Float64(0.0 - Float64(sqrt(2.0) / B_m)) * sqrt(Float64(F * Float64(C + hypot(C, 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)
	t_0 = ((A * (4.0 * C)) - (B_m * B_m)) / sqrt((A + (C + hypot(B_m, (A - C)))));
	tmp = 0.0;
	if (B_m <= 3.6e+43)
		tmp = sqrt((((B_m * B_m) + (-4.0 * (A * C))) * (2.0 * F))) / t_0;
	elseif (B_m <= 1.05e+153)
		tmp = (sqrt(F) * (B_m * sqrt(2.0))) / t_0;
	elseif (B_m <= 1.2e+224)
		tmp = (0.0 - (sqrt(2.0) / B_m)) * sqrt((F * (C + hypot(C, 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_] := Block[{t$95$0 = N[(N[(N[(A * N[(4.0 * C), $MachinePrecision]), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 3.6e+43], N[(N[Sqrt[N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[B$95$m, 1.05e+153], N[(N[(N[Sqrt[F], $MachinePrecision] * N[(B$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[B$95$m, 1.2e+224], N[(N[(0.0 - N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(F * N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision] * N[(0.0 - N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 3.6000000000000001e43

   1. Initial program 17.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 24.8%

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

      1. pow1/2 N/A

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

      2. unpow-prod-down N/A

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

      3. associate-/l* N/A

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

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

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

   6. Applied egg-rr 33.6%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

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

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

   8. Applied egg-rr 33.6%

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

   if 3.6000000000000001e43 < B < 1.05000000000000008e153

   1. Initial program 30.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 38.9%

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

      1. pow1/2 N/A

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

      2. unpow-prod-down N/A

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

      3. associate-/l* N/A

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

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

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

   6. Applied egg-rr 50.4%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

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

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

   8. Applied egg-rr 50.4%

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

   9. Taylor expanded in B around inf

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

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

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

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

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

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

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

       4. sqrt-lowering-sqrt.f64 66.3%

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

   11. Simplified 66.3%

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

   if 1.05000000000000008e153 < B < 1.2e224

   1. Initial program 8.9%

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

   3. Taylor expanded in A around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. +-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 55.7%

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

   5. Simplified 55.7%

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

   if 1.2e224 < B

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

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

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

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

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

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

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

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

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

      6. sqrt-lowering-sqrt.f64 67.0%

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

   5. Simplified 67.0%

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

4. Final simplification 40.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.6 \cdot 10^{+43}:\\ \;\;\;\;\frac{\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)}}{\frac{A \cdot \left(4 \cdot C\right) - B \cdot B}{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}}}\\ \mathbf{elif}\;B \leq 1.05 \cdot 10^{+153}:\\ \;\;\;\;\frac{\sqrt{F} \cdot \left(B \cdot \sqrt{2}\right)}{\frac{A \cdot \left(4 \cdot C\right) - B \cdot B}{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}}}\\ \mathbf{elif}\;B \leq 1.2 \cdot 10^{+224}:\\ \;\;\;\;\left(0 - \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(0 - \sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 39.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(B\_m, A - C\right)\\ \mathbf{if}\;B\_m \leq 6.5 \cdot 10^{-192}:\\ \;\;\;\;\sqrt{-8 \cdot \left(A \cdot \left(C \cdot F\right)\right)} \cdot \left(0.25 \cdot \left(\frac{\sqrt{2}}{A} \cdot \sqrt{\frac{1}{C}}\right)\right)\\ \mathbf{elif}\;B\_m \leq 1.95 \cdot 10^{-92}:\\ \;\;\;\;\frac{\sqrt{\left(C + \left(A + t\_0\right)\right) \cdot \left(\left(B\_m \cdot B\_m + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)\right)}}{4 \cdot \left(A \cdot C\right) - B\_m \cdot B\_m}\\ \mathbf{elif}\;B\_m \leq 7 \cdot 10^{+152}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(A + \left(C + t\_0\right)\right)} \cdot \left(B\_m \cdot \sqrt{F}\right)}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{elif}\;B\_m \leq 7.5 \cdot 10^{+222}:\\ \;\;\;\;\left(0 - \frac{\sqrt{2}}{B\_m}\right) \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B\_m}} \cdot \left(0 - \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (hypot B_m (- A C))))
   (if (<= B_m 6.5e-192)
     (*
      (sqrt (* -8.0 (* A (* C F))))
      (* 0.25 (* (/ (sqrt 2.0) A) (sqrt (/ 1.0 C)))))
     (if (<= B_m 1.95e-92)
       (/
        (sqrt
         (* (+ C (+ A t_0)) (* (+ (* B_m B_m) (* -4.0 (* A C))) (* 2.0 F))))
        (- (* 4.0 (* A C)) (* B_m B_m)))
       (if (<= B_m 7e+152)
         (/
          (* (sqrt (* 2.0 (+ A (+ C t_0)))) (* B_m (sqrt F)))
          (- (* (* 4.0 A) C) (* B_m B_m)))
         (if (<= B_m 7.5e+222)
           (* (- 0.0 (/ (sqrt 2.0) B_m)) (sqrt (* F (+ C (hypot C 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 t_0 = hypot(B_m, (A - C));
	double tmp;
	if (B_m <= 6.5e-192) {
		tmp = sqrt((-8.0 * (A * (C * F)))) * (0.25 * ((sqrt(2.0) / A) * sqrt((1.0 / C))));
	} else if (B_m <= 1.95e-92) {
		tmp = sqrt(((C + (A + t_0)) * (((B_m * B_m) + (-4.0 * (A * C))) * (2.0 * F)))) / ((4.0 * (A * C)) - (B_m * B_m));
	} else if (B_m <= 7e+152) {
		tmp = (sqrt((2.0 * (A + (C + t_0)))) * (B_m * sqrt(F))) / (((4.0 * A) * C) - (B_m * B_m));
	} else if (B_m <= 7.5e+222) {
		tmp = (0.0 - (sqrt(2.0) / B_m)) * sqrt((F * (C + hypot(C, B_m))));
	} 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 t_0 = Math.hypot(B_m, (A - C));
	double tmp;
	if (B_m <= 6.5e-192) {
		tmp = Math.sqrt((-8.0 * (A * (C * F)))) * (0.25 * ((Math.sqrt(2.0) / A) * Math.sqrt((1.0 / C))));
	} else if (B_m <= 1.95e-92) {
		tmp = Math.sqrt(((C + (A + t_0)) * (((B_m * B_m) + (-4.0 * (A * C))) * (2.0 * F)))) / ((4.0 * (A * C)) - (B_m * B_m));
	} else if (B_m <= 7e+152) {
		tmp = (Math.sqrt((2.0 * (A + (C + t_0)))) * (B_m * Math.sqrt(F))) / (((4.0 * A) * C) - (B_m * B_m));
	} else if (B_m <= 7.5e+222) {
		tmp = (0.0 - (Math.sqrt(2.0) / B_m)) * Math.sqrt((F * (C + Math.hypot(C, 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):
	t_0 = math.hypot(B_m, (A - C))
	tmp = 0
	if B_m <= 6.5e-192:
		tmp = math.sqrt((-8.0 * (A * (C * F)))) * (0.25 * ((math.sqrt(2.0) / A) * math.sqrt((1.0 / C))))
	elif B_m <= 1.95e-92:
		tmp = math.sqrt(((C + (A + t_0)) * (((B_m * B_m) + (-4.0 * (A * C))) * (2.0 * F)))) / ((4.0 * (A * C)) - (B_m * B_m))
	elif B_m <= 7e+152:
		tmp = (math.sqrt((2.0 * (A + (C + t_0)))) * (B_m * math.sqrt(F))) / (((4.0 * A) * C) - (B_m * B_m))
	elif B_m <= 7.5e+222:
		tmp = (0.0 - (math.sqrt(2.0) / B_m)) * math.sqrt((F * (C + math.hypot(C, 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)
	t_0 = hypot(B_m, Float64(A - C))
	tmp = 0.0
	if (B_m <= 6.5e-192)
		tmp = Float64(sqrt(Float64(-8.0 * Float64(A * Float64(C * F)))) * Float64(0.25 * Float64(Float64(sqrt(2.0) / A) * sqrt(Float64(1.0 / C)))));
	elseif (B_m <= 1.95e-92)
		tmp = Float64(sqrt(Float64(Float64(C + Float64(A + t_0)) * Float64(Float64(Float64(B_m * B_m) + Float64(-4.0 * Float64(A * C))) * Float64(2.0 * F)))) / Float64(Float64(4.0 * Float64(A * C)) - Float64(B_m * B_m)));
	elseif (B_m <= 7e+152)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(A + Float64(C + t_0)))) * Float64(B_m * sqrt(F))) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m)));
	elseif (B_m <= 7.5e+222)
		tmp = Float64(Float64(0.0 - Float64(sqrt(2.0) / B_m)) * sqrt(Float64(F * Float64(C + hypot(C, 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)
	t_0 = hypot(B_m, (A - C));
	tmp = 0.0;
	if (B_m <= 6.5e-192)
		tmp = sqrt((-8.0 * (A * (C * F)))) * (0.25 * ((sqrt(2.0) / A) * sqrt((1.0 / C))));
	elseif (B_m <= 1.95e-92)
		tmp = sqrt(((C + (A + t_0)) * (((B_m * B_m) + (-4.0 * (A * C))) * (2.0 * F)))) / ((4.0 * (A * C)) - (B_m * B_m));
	elseif (B_m <= 7e+152)
		tmp = (sqrt((2.0 * (A + (C + t_0)))) * (B_m * sqrt(F))) / (((4.0 * A) * C) - (B_m * B_m));
	elseif (B_m <= 7.5e+222)
		tmp = (0.0 - (sqrt(2.0) / B_m)) * sqrt((F * (C + hypot(C, 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_] := Block[{t$95$0 = N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[B$95$m, 6.5e-192], N[(N[Sqrt[N[(-8.0 * N[(A * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(0.25 * N[(N[(N[Sqrt[2.0], $MachinePrecision] / A), $MachinePrecision] * N[Sqrt[N[(1.0 / C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 1.95e-92], N[(N[Sqrt[N[(N[(C + N[(A + t$95$0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 7e+152], N[(N[(N[Sqrt[N[(2.0 * N[(A + N[(C + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(B$95$m * N[Sqrt[F], $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, 7.5e+222], N[(N[(0.0 - N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(F * N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision] * N[(0.0 - N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if B < 6.49999999999999966e-192

   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 23.9%

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

      1. pow1/2 N/A

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

      2. unpow-prod-down N/A

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

      3. associate-/l* N/A

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

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

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

   6. Applied egg-rr 32.1%

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

   7. Taylor expanded in A around -inf

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

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

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

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

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

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

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

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

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

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

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

      6. /-lowering-/.f64 13.7%

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

   9. Simplified 13.7%

      \[\leadsto \sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)} \cdot \color{blue}{\left(0.25 \cdot \left(\frac{\sqrt{2}}{A} \cdot \sqrt{\frac{1}{C}}\right)\right)} \]

   10. Taylor expanded in B around 0

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

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-8, \left(A \cdot \left(C \cdot F\right)\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), A\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, C\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-8, \mathsf{*.f64}\left(A, \left(C \cdot F\right)\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), A\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, C\right)\right)\right)\right)\right) \]

       3. *-lowering-*.f64 15.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-8, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, F\right)\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), A\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, C\right)\right)\right)\right)\right) \]

   12. Simplified 15.9%

       \[\leadsto \sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot F\right)\right)}} \cdot \left(0.25 \cdot \left(\frac{\sqrt{2}}{A} \cdot \sqrt{\frac{1}{C}}\right)\right) \]

   if 6.49999999999999966e-192 < B < 1.9499999999999998e-92

   1. Initial program 29.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 45.3%

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

      1. pow1/2 N/A

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

      2. unpow-prod-down N/A

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

      3. associate-/l* N/A

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

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

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

   6. Applied egg-rr 46.9%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

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

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

   8. Applied egg-rr 47.1%

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

      1. pow1/2 N/A

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

      2. associate-*r* N/A

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

      3. unpow-prod-down N/A

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

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

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

      5. pow1/2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. pow1/2 N/A

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

      13. sqrt-lowering-sqrt.f64 24.7%

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

   10. Applied egg-rr 24.7%

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

       1. associate-/r/ N/A

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

       2. sqrt-unprod N/A

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

       3. associate-*r* N/A

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

       4. associate-*l/ N/A

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

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

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

   12. Applied egg-rr 45.2%

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

   if 1.9499999999999998e-92 < B < 6.99999999999999963e152

   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 26.9%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   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 41.3%

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

   7. Taylor expanded in B around inf

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

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

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

      2. sqrt-lowering-sqrt.f64 45.2%

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

   9. Simplified 45.2%

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

   if 6.99999999999999963e152 < B < 7.50000000000000003e222

   1. Initial program 8.9%

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

   3. Taylor expanded in A around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. +-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 55.7%

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

   5. Simplified 55.7%

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

   if 7.50000000000000003e222 < B

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

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

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

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

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

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

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

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

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

      6. sqrt-lowering-sqrt.f64 67.0%

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

   5. Simplified 67.0%

      \[\leadsto \color{blue}{-\sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 29.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 6.5 \cdot 10^{-192}:\\ \;\;\;\;\sqrt{-8 \cdot \left(A \cdot \left(C \cdot F\right)\right)} \cdot \left(0.25 \cdot \left(\frac{\sqrt{2}}{A} \cdot \sqrt{\frac{1}{C}}\right)\right)\\ \mathbf{elif}\;B \leq 1.95 \cdot 10^{-92}:\\ \;\;\;\;\frac{\sqrt{\left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)\right)}}{4 \cdot \left(A \cdot C\right) - B \cdot B}\\ \mathbf{elif}\;B \leq 7 \cdot 10^{+152}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(B \cdot \sqrt{F}\right)}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;B \leq 7.5 \cdot 10^{+222}:\\ \;\;\;\;\left(0 - \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(0 - \sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 44.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 8.5 \cdot 10^{+140}:\\ \;\;\;\;\frac{\sqrt{\left(B\_m \cdot B\_m + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)}}{\frac{A \cdot \left(4 \cdot C\right) - B\_m \cdot B\_m}{\sqrt{A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)}}}\\ \mathbf{elif}\;B\_m \leq 2.3 \cdot 10^{+222}:\\ \;\;\;\;\left(0 - \frac{\sqrt{2}}{B\_m}\right) \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B\_m}} \cdot \left(0 - \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 8.5e+140)
   (/
    (sqrt (* (+ (* B_m B_m) (* -4.0 (* A C))) (* 2.0 F)))
    (/ (- (* A (* 4.0 C)) (* B_m B_m)) (sqrt (+ A (+ C (hypot B_m (- A C)))))))
   (if (<= B_m 2.3e+222)
     (* (- 0.0 (/ (sqrt 2.0) B_m)) (sqrt (* F (+ C (hypot C 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 <= 8.5e+140) {
		tmp = sqrt((((B_m * B_m) + (-4.0 * (A * C))) * (2.0 * F))) / (((A * (4.0 * C)) - (B_m * B_m)) / sqrt((A + (C + hypot(B_m, (A - C))))));
	} else if (B_m <= 2.3e+222) {
		tmp = (0.0 - (sqrt(2.0) / B_m)) * sqrt((F * (C + hypot(C, B_m))));
	} else {
		tmp = sqrt((F / B_m)) * (0.0 - sqrt(2.0));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 8.5e+140) {
		tmp = Math.sqrt((((B_m * B_m) + (-4.0 * (A * C))) * (2.0 * F))) / (((A * (4.0 * C)) - (B_m * B_m)) / Math.sqrt((A + (C + Math.hypot(B_m, (A - C))))));
	} else if (B_m <= 2.3e+222) {
		tmp = (0.0 - (Math.sqrt(2.0) / B_m)) * Math.sqrt((F * (C + Math.hypot(C, 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 <= 8.5e+140:
		tmp = math.sqrt((((B_m * B_m) + (-4.0 * (A * C))) * (2.0 * F))) / (((A * (4.0 * C)) - (B_m * B_m)) / math.sqrt((A + (C + math.hypot(B_m, (A - C))))))
	elif B_m <= 2.3e+222:
		tmp = (0.0 - (math.sqrt(2.0) / B_m)) * math.sqrt((F * (C + math.hypot(C, 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 <= 8.5e+140)
		tmp = Float64(sqrt(Float64(Float64(Float64(B_m * B_m) + Float64(-4.0 * Float64(A * C))) * Float64(2.0 * F))) / Float64(Float64(Float64(A * Float64(4.0 * C)) - Float64(B_m * B_m)) / sqrt(Float64(A + Float64(C + hypot(B_m, Float64(A - C)))))));
	elseif (B_m <= 2.3e+222)
		tmp = Float64(Float64(0.0 - Float64(sqrt(2.0) / B_m)) * sqrt(Float64(F * Float64(C + hypot(C, 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 <= 8.5e+140)
		tmp = sqrt((((B_m * B_m) + (-4.0 * (A * C))) * (2.0 * F))) / (((A * (4.0 * C)) - (B_m * B_m)) / sqrt((A + (C + hypot(B_m, (A - C))))));
	elseif (B_m <= 2.3e+222)
		tmp = (0.0 - (sqrt(2.0) / B_m)) * sqrt((F * (C + hypot(C, 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, 8.5e+140], N[(N[Sqrt[N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(A * N[(4.0 * C), $MachinePrecision]), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 2.3e+222], N[(N[(0.0 - N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(F * N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision] * N[(0.0 - N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 8.4999999999999996e140

   1. Initial program 19.4%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 26.4%

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

      1. pow1/2 N/A

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

      2. unpow-prod-down N/A

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

      3. associate-/l* N/A

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

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

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

   6. Applied egg-rr 35.1%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

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

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

   8. Applied egg-rr 35.2%

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

   if 8.4999999999999996e140 < B < 2.30000000000000011e222

   1. Initial program 8.4%

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

   3. Taylor expanded in 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 59.2%

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

   5. Simplified 59.2%

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

   if 2.30000000000000011e222 < B

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

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

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

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

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

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

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

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

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

      6. sqrt-lowering-sqrt.f64 67.0%

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

   5. Simplified 67.0%

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

4. Final simplification 38.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 8.5 \cdot 10^{+140}:\\ \;\;\;\;\frac{\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)}}{\frac{A \cdot \left(4 \cdot C\right) - B \cdot B}{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}}}\\ \mathbf{elif}\;B \leq 2.3 \cdot 10^{+222}:\\ \;\;\;\;\left(0 - \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(0 - \sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 41.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)\\ \mathbf{if}\;B\_m \leq 1.55 \cdot 10^{-99}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)}}{\frac{A \cdot \left(4 \cdot C\right) - B\_m \cdot B\_m}{\sqrt{t\_0}}}\\ \mathbf{elif}\;B\_m \leq 1.05 \cdot 10^{+153}:\\ \;\;\;\;\frac{\sqrt{2 \cdot t\_0} \cdot \left(B\_m \cdot \sqrt{F}\right)}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{elif}\;B\_m \leq 8.2 \cdot 10^{+222}:\\ \;\;\;\;\left(0 - \frac{\sqrt{2}}{B\_m}\right) \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B\_m}} \cdot \left(0 - \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (+ A (+ C (hypot B_m (- A C))))))
   (if (<= B_m 1.55e-99)
     (/
      (sqrt (* (* -4.0 (* A C)) (* 2.0 F)))
      (/ (- (* A (* 4.0 C)) (* B_m B_m)) (sqrt t_0)))
     (if (<= B_m 1.05e+153)
       (/
        (* (sqrt (* 2.0 t_0)) (* B_m (sqrt F)))
        (- (* (* 4.0 A) C) (* B_m B_m)))
       (if (<= B_m 8.2e+222)
         (* (- 0.0 (/ (sqrt 2.0) B_m)) (sqrt (* F (+ C (hypot C 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 t_0 = A + (C + hypot(B_m, (A - C)));
	double tmp;
	if (B_m <= 1.55e-99) {
		tmp = sqrt(((-4.0 * (A * C)) * (2.0 * F))) / (((A * (4.0 * C)) - (B_m * B_m)) / sqrt(t_0));
	} else if (B_m <= 1.05e+153) {
		tmp = (sqrt((2.0 * t_0)) * (B_m * sqrt(F))) / (((4.0 * A) * C) - (B_m * B_m));
	} else if (B_m <= 8.2e+222) {
		tmp = (0.0 - (sqrt(2.0) / B_m)) * sqrt((F * (C + hypot(C, B_m))));
	} 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 t_0 = A + (C + Math.hypot(B_m, (A - C)));
	double tmp;
	if (B_m <= 1.55e-99) {
		tmp = Math.sqrt(((-4.0 * (A * C)) * (2.0 * F))) / (((A * (4.0 * C)) - (B_m * B_m)) / Math.sqrt(t_0));
	} else if (B_m <= 1.05e+153) {
		tmp = (Math.sqrt((2.0 * t_0)) * (B_m * Math.sqrt(F))) / (((4.0 * A) * C) - (B_m * B_m));
	} else if (B_m <= 8.2e+222) {
		tmp = (0.0 - (Math.sqrt(2.0) / B_m)) * Math.sqrt((F * (C + Math.hypot(C, 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):
	t_0 = A + (C + math.hypot(B_m, (A - C)))
	tmp = 0
	if B_m <= 1.55e-99:
		tmp = math.sqrt(((-4.0 * (A * C)) * (2.0 * F))) / (((A * (4.0 * C)) - (B_m * B_m)) / math.sqrt(t_0))
	elif B_m <= 1.05e+153:
		tmp = (math.sqrt((2.0 * t_0)) * (B_m * math.sqrt(F))) / (((4.0 * A) * C) - (B_m * B_m))
	elif B_m <= 8.2e+222:
		tmp = (0.0 - (math.sqrt(2.0) / B_m)) * math.sqrt((F * (C + math.hypot(C, 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)
	t_0 = Float64(A + Float64(C + hypot(B_m, Float64(A - C))))
	tmp = 0.0
	if (B_m <= 1.55e-99)
		tmp = Float64(sqrt(Float64(Float64(-4.0 * Float64(A * C)) * Float64(2.0 * F))) / Float64(Float64(Float64(A * Float64(4.0 * C)) - Float64(B_m * B_m)) / sqrt(t_0)));
	elseif (B_m <= 1.05e+153)
		tmp = Float64(Float64(sqrt(Float64(2.0 * t_0)) * Float64(B_m * sqrt(F))) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m)));
	elseif (B_m <= 8.2e+222)
		tmp = Float64(Float64(0.0 - Float64(sqrt(2.0) / B_m)) * sqrt(Float64(F * Float64(C + hypot(C, 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)
	t_0 = A + (C + hypot(B_m, (A - C)));
	tmp = 0.0;
	if (B_m <= 1.55e-99)
		tmp = sqrt(((-4.0 * (A * C)) * (2.0 * F))) / (((A * (4.0 * C)) - (B_m * B_m)) / sqrt(t_0));
	elseif (B_m <= 1.05e+153)
		tmp = (sqrt((2.0 * t_0)) * (B_m * sqrt(F))) / (((4.0 * A) * C) - (B_m * B_m));
	elseif (B_m <= 8.2e+222)
		tmp = (0.0 - (sqrt(2.0) / B_m)) * sqrt((F * (C + hypot(C, 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_] := Block[{t$95$0 = N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 1.55e-99], N[(N[Sqrt[N[(N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(A * N[(4.0 * C), $MachinePrecision]), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 1.05e+153], N[(N[(N[Sqrt[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision] * N[(B$95$m * N[Sqrt[F], $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, 8.2e+222], N[(N[(0.0 - N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(F * N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision] * N[(0.0 - N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 1.5499999999999999e-99

   1. Initial program 18.3%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 26.3%

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

      1. pow1/2 N/A

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

      2. unpow-prod-down N/A

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

      3. associate-/l* N/A

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

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

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

   6. Applied egg-rr 33.8%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

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

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

   8. Applied egg-rr 33.9%

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

   9. Taylor expanded in B around 0

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

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

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

       2. *-lowering-*.f64 21.8%

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

   11. Simplified 21.8%

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

   if 1.5499999999999999e-99 < B < 1.05000000000000008e153

   1. Initial program 22.5%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 26.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing
   5. 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 40.6%

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

   7. Taylor expanded in B around inf

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

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

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

      2. sqrt-lowering-sqrt.f64 44.4%

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

   9. Simplified 44.4%

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

   if 1.05000000000000008e153 < B < 8.19999999999999974e222

   1. Initial program 8.9%

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

   3. Taylor expanded in A around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. +-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 55.7%

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

   5. Simplified 55.7%

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

   if 8.19999999999999974e222 < B

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

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

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

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

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

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

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

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

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

      6. sqrt-lowering-sqrt.f64 67.0%

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

   5. Simplified 67.0%

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

4. Final simplification 31.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.55 \cdot 10^{-99}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)}}{\frac{A \cdot \left(4 \cdot C\right) - B \cdot B}{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}}}\\ \mathbf{elif}\;B \leq 1.05 \cdot 10^{+153}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(B \cdot \sqrt{F}\right)}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;B \leq 8.2 \cdot 10^{+222}:\\ \;\;\;\;\left(0 - \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(0 - \sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 40.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)\\ \mathbf{if}\;B\_m \leq 1.05 \cdot 10^{-108}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot F\right)\right)}}{\frac{A \cdot \left(4 \cdot C\right) - B\_m \cdot B\_m}{\sqrt{t\_0}}}\\ \mathbf{elif}\;B\_m \leq 3.65 \cdot 10^{+152}:\\ \;\;\;\;\frac{\sqrt{2 \cdot t\_0} \cdot \left(B\_m \cdot \sqrt{F}\right)}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{elif}\;B\_m \leq 3.6 \cdot 10^{+224}:\\ \;\;\;\;\left(0 - \frac{\sqrt{2}}{B\_m}\right) \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B\_m}} \cdot \left(0 - \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (+ A (+ C (hypot B_m (- A C))))))
   (if (<= B_m 1.05e-108)
     (/
      (sqrt (* -8.0 (* A (* C F))))
      (/ (- (* A (* 4.0 C)) (* B_m B_m)) (sqrt t_0)))
     (if (<= B_m 3.65e+152)
       (/
        (* (sqrt (* 2.0 t_0)) (* B_m (sqrt F)))
        (- (* (* 4.0 A) C) (* B_m B_m)))
       (if (<= B_m 3.6e+224)
         (* (- 0.0 (/ (sqrt 2.0) B_m)) (sqrt (* F (+ C (hypot C 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 t_0 = A + (C + hypot(B_m, (A - C)));
	double tmp;
	if (B_m <= 1.05e-108) {
		tmp = sqrt((-8.0 * (A * (C * F)))) / (((A * (4.0 * C)) - (B_m * B_m)) / sqrt(t_0));
	} else if (B_m <= 3.65e+152) {
		tmp = (sqrt((2.0 * t_0)) * (B_m * sqrt(F))) / (((4.0 * A) * C) - (B_m * B_m));
	} else if (B_m <= 3.6e+224) {
		tmp = (0.0 - (sqrt(2.0) / B_m)) * sqrt((F * (C + hypot(C, B_m))));
	} 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 t_0 = A + (C + Math.hypot(B_m, (A - C)));
	double tmp;
	if (B_m <= 1.05e-108) {
		tmp = Math.sqrt((-8.0 * (A * (C * F)))) / (((A * (4.0 * C)) - (B_m * B_m)) / Math.sqrt(t_0));
	} else if (B_m <= 3.65e+152) {
		tmp = (Math.sqrt((2.0 * t_0)) * (B_m * Math.sqrt(F))) / (((4.0 * A) * C) - (B_m * B_m));
	} else if (B_m <= 3.6e+224) {
		tmp = (0.0 - (Math.sqrt(2.0) / B_m)) * Math.sqrt((F * (C + Math.hypot(C, 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):
	t_0 = A + (C + math.hypot(B_m, (A - C)))
	tmp = 0
	if B_m <= 1.05e-108:
		tmp = math.sqrt((-8.0 * (A * (C * F)))) / (((A * (4.0 * C)) - (B_m * B_m)) / math.sqrt(t_0))
	elif B_m <= 3.65e+152:
		tmp = (math.sqrt((2.0 * t_0)) * (B_m * math.sqrt(F))) / (((4.0 * A) * C) - (B_m * B_m))
	elif B_m <= 3.6e+224:
		tmp = (0.0 - (math.sqrt(2.0) / B_m)) * math.sqrt((F * (C + math.hypot(C, 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)
	t_0 = Float64(A + Float64(C + hypot(B_m, Float64(A - C))))
	tmp = 0.0
	if (B_m <= 1.05e-108)
		tmp = Float64(sqrt(Float64(-8.0 * Float64(A * Float64(C * F)))) / Float64(Float64(Float64(A * Float64(4.0 * C)) - Float64(B_m * B_m)) / sqrt(t_0)));
	elseif (B_m <= 3.65e+152)
		tmp = Float64(Float64(sqrt(Float64(2.0 * t_0)) * Float64(B_m * sqrt(F))) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m)));
	elseif (B_m <= 3.6e+224)
		tmp = Float64(Float64(0.0 - Float64(sqrt(2.0) / B_m)) * sqrt(Float64(F * Float64(C + hypot(C, 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)
	t_0 = A + (C + hypot(B_m, (A - C)));
	tmp = 0.0;
	if (B_m <= 1.05e-108)
		tmp = sqrt((-8.0 * (A * (C * F)))) / (((A * (4.0 * C)) - (B_m * B_m)) / sqrt(t_0));
	elseif (B_m <= 3.65e+152)
		tmp = (sqrt((2.0 * t_0)) * (B_m * sqrt(F))) / (((4.0 * A) * C) - (B_m * B_m));
	elseif (B_m <= 3.6e+224)
		tmp = (0.0 - (sqrt(2.0) / B_m)) * sqrt((F * (C + hypot(C, 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_] := Block[{t$95$0 = N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 1.05e-108], N[(N[Sqrt[N[(-8.0 * N[(A * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(A * N[(4.0 * C), $MachinePrecision]), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 3.65e+152], N[(N[(N[Sqrt[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision] * N[(B$95$m * N[Sqrt[F], $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, 3.6e+224], N[(N[(0.0 - N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(F * N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision] * N[(0.0 - N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 1.05e-108

   1. Initial program 17.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 25.4%

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

      1. pow1/2 N/A

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

      2. unpow-prod-down N/A

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

      3. associate-/l* N/A

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

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

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

   6. Applied egg-rr 33.1%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

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

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

   8. Applied egg-rr 33.1%

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

   9. Taylor expanded in B around 0

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

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

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

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

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

       3. *-lowering-*.f64 18.6%

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

   11. Simplified 18.6%

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

   if 1.05e-108 < B < 3.6500000000000002e152

   1. Initial program 23.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 29.0%

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

      1. 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 42.7%

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

   7. Taylor expanded in B around inf

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

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

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

      2. sqrt-lowering-sqrt.f64 42.9%

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

   9. Simplified 42.9%

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

   if 3.6500000000000002e152 < B < 3.6e224

   1. Initial program 8.9%

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

   3. Taylor expanded in A around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. +-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 55.7%

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

   5. Simplified 55.7%

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

   if 3.6e224 < B

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

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

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

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

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

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

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

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

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

      6. sqrt-lowering-sqrt.f64 67.0%

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

   5. Simplified 67.0%

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

4. Final simplification 29.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.05 \cdot 10^{-108}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot F\right)\right)}}{\frac{A \cdot \left(4 \cdot C\right) - B \cdot B}{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}}}\\ \mathbf{elif}\;B \leq 3.65 \cdot 10^{+152}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(B \cdot \sqrt{F}\right)}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;B \leq 3.6 \cdot 10^{+224}:\\ \;\;\;\;\left(0 - \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(0 - \sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 41.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;F \leq 4.45 \cdot 10^{-305}:\\ \;\;\;\;\frac{\sqrt{\left(C + \left(A + \mathsf{hypot}\left(B\_m, A - C\right)\right)\right) \cdot \left(\left(B\_m \cdot B\_m + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)\right)}}{4 \cdot \left(A \cdot C\right) - B\_m \cdot B\_m}\\ \mathbf{elif}\;F \leq 2300000:\\ \;\;\;\;\left(0 - \frac{\sqrt{2}}{B\_m}\right) \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B\_m}} \cdot \left(0 - \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F 4.45e-305)
   (/
    (sqrt
     (*
      (+ C (+ A (hypot B_m (- A C))))
      (* (+ (* B_m B_m) (* -4.0 (* A C))) (* 2.0 F))))
    (- (* 4.0 (* A C)) (* B_m B_m)))
   (if (<= F 2300000.0)
     (* (- 0.0 (/ (sqrt 2.0) B_m)) (sqrt (* F (+ C (hypot C 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 (F <= 4.45e-305) {
		tmp = sqrt(((C + (A + hypot(B_m, (A - C)))) * (((B_m * B_m) + (-4.0 * (A * C))) * (2.0 * F)))) / ((4.0 * (A * C)) - (B_m * B_m));
	} else if (F <= 2300000.0) {
		tmp = (0.0 - (sqrt(2.0) / B_m)) * sqrt((F * (C + hypot(C, B_m))));
	} 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 <= 4.45e-305) {
		tmp = Math.sqrt(((C + (A + Math.hypot(B_m, (A - C)))) * (((B_m * B_m) + (-4.0 * (A * C))) * (2.0 * F)))) / ((4.0 * (A * C)) - (B_m * B_m));
	} else if (F <= 2300000.0) {
		tmp = (0.0 - (Math.sqrt(2.0) / B_m)) * Math.sqrt((F * (C + Math.hypot(C, 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 F <= 4.45e-305:
		tmp = math.sqrt(((C + (A + math.hypot(B_m, (A - C)))) * (((B_m * B_m) + (-4.0 * (A * C))) * (2.0 * F)))) / ((4.0 * (A * C)) - (B_m * B_m))
	elif F <= 2300000.0:
		tmp = (0.0 - (math.sqrt(2.0) / B_m)) * math.sqrt((F * (C + math.hypot(C, 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 (F <= 4.45e-305)
		tmp = Float64(sqrt(Float64(Float64(C + Float64(A + hypot(B_m, Float64(A - C)))) * Float64(Float64(Float64(B_m * B_m) + Float64(-4.0 * Float64(A * C))) * Float64(2.0 * F)))) / Float64(Float64(4.0 * Float64(A * C)) - Float64(B_m * B_m)));
	elseif (F <= 2300000.0)
		tmp = Float64(Float64(0.0 - Float64(sqrt(2.0) / B_m)) * sqrt(Float64(F * Float64(C + hypot(C, 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 (F <= 4.45e-305)
		tmp = sqrt(((C + (A + hypot(B_m, (A - C)))) * (((B_m * B_m) + (-4.0 * (A * C))) * (2.0 * F)))) / ((4.0 * (A * C)) - (B_m * B_m));
	elseif (F <= 2300000.0)
		tmp = (0.0 - (sqrt(2.0) / B_m)) * sqrt((F * (C + hypot(C, 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[F, 4.45e-305], N[(N[Sqrt[N[(N[(C + N[(A + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2300000.0], N[(N[(0.0 - N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(F * N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision] * N[(0.0 - N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < 4.4499999999999998e-305

   1. Initial program 24.3%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 37.5%

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

      1. pow1/2 N/A

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

      2. unpow-prod-down N/A

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

      3. associate-/l* N/A

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

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

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

   6. Applied egg-rr 53.4%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

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

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

   8. Applied egg-rr 53.4%

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

      1. pow1/2 N/A

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

      2. associate-*r* N/A

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

      3. unpow-prod-down N/A

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

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

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

      5. pow1/2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. pow1/2 N/A

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

      13. sqrt-lowering-sqrt.f64 3.3%

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

   10. Applied egg-rr 3.3%

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

       1. associate-/r/ N/A

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

       2. sqrt-unprod N/A

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

       3. associate-*r* N/A

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

       4. associate-*l/ N/A

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

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

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

   12. Applied egg-rr 37.5%

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

   if 4.4499999999999998e-305 < F < 2.3e6

   1. Initial program 18.7%

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

   3. Taylor expanded in 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%

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

   if 2.3e6 < F

   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. 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 21.0%

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

   5. Simplified 21.0%

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

4. Final simplification 23.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 4.45 \cdot 10^{-305}:\\ \;\;\;\;\frac{\sqrt{\left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)\right)}}{4 \cdot \left(A \cdot C\right) - B \cdot B}\\ \mathbf{elif}\;F \leq 2300000:\\ \;\;\;\;\left(0 - \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(0 - \sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 41.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;F \leq 4.4 \cdot 10^{-305}:\\ \;\;\;\;\frac{\sqrt{\left(C + \left(A + \mathsf{hypot}\left(B\_m, A - C\right)\right)\right) \cdot \left(\left(B\_m \cdot B\_m + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)\right)}}{4 \cdot \left(A \cdot C\right) - B\_m \cdot B\_m}\\ \mathbf{elif}\;F \leq 9.8 \cdot 10^{-11}:\\ \;\;\;\;\left(0 - \frac{\sqrt{2}}{B\_m}\right) \cdot \sqrt{F \cdot \left(A + \mathsf{hypot}\left(B\_m, A\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B\_m}} \cdot \left(0 - \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F 4.4e-305)
   (/
    (sqrt
     (*
      (+ C (+ A (hypot B_m (- A C))))
      (* (+ (* B_m B_m) (* -4.0 (* A C))) (* 2.0 F))))
    (- (* 4.0 (* A C)) (* B_m B_m)))
   (if (<= F 9.8e-11)
     (* (- 0.0 (/ (sqrt 2.0) B_m)) (sqrt (* F (+ A (hypot B_m A)))))
     (* (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 <= 4.4e-305) {
		tmp = sqrt(((C + (A + hypot(B_m, (A - C)))) * (((B_m * B_m) + (-4.0 * (A * C))) * (2.0 * F)))) / ((4.0 * (A * C)) - (B_m * B_m));
	} else if (F <= 9.8e-11) {
		tmp = (0.0 - (sqrt(2.0) / B_m)) * sqrt((F * (A + hypot(B_m, A))));
	} 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 <= 4.4e-305) {
		tmp = Math.sqrt(((C + (A + Math.hypot(B_m, (A - C)))) * (((B_m * B_m) + (-4.0 * (A * C))) * (2.0 * F)))) / ((4.0 * (A * C)) - (B_m * B_m));
	} else if (F <= 9.8e-11) {
		tmp = (0.0 - (Math.sqrt(2.0) / B_m)) * Math.sqrt((F * (A + Math.hypot(B_m, A))));
	} 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 <= 4.4e-305:
		tmp = math.sqrt(((C + (A + math.hypot(B_m, (A - C)))) * (((B_m * B_m) + (-4.0 * (A * C))) * (2.0 * F)))) / ((4.0 * (A * C)) - (B_m * B_m))
	elif F <= 9.8e-11:
		tmp = (0.0 - (math.sqrt(2.0) / B_m)) * math.sqrt((F * (A + math.hypot(B_m, A))))
	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 <= 4.4e-305)
		tmp = Float64(sqrt(Float64(Float64(C + Float64(A + hypot(B_m, Float64(A - C)))) * Float64(Float64(Float64(B_m * B_m) + Float64(-4.0 * Float64(A * C))) * Float64(2.0 * F)))) / Float64(Float64(4.0 * Float64(A * C)) - Float64(B_m * B_m)));
	elseif (F <= 9.8e-11)
		tmp = Float64(Float64(0.0 - Float64(sqrt(2.0) / B_m)) * sqrt(Float64(F * Float64(A + hypot(B_m, A)))));
	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 <= 4.4e-305)
		tmp = sqrt(((C + (A + hypot(B_m, (A - C)))) * (((B_m * B_m) + (-4.0 * (A * C))) * (2.0 * F)))) / ((4.0 * (A * C)) - (B_m * B_m));
	elseif (F <= 9.8e-11)
		tmp = (0.0 - (sqrt(2.0) / B_m)) * sqrt((F * (A + hypot(B_m, A))));
	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, 4.4e-305], N[(N[Sqrt[N[(N[(C + N[(A + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 9.8e-11], N[(N[(0.0 - N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(F * N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision] * N[(0.0 - N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < 4.39999999999999993e-305

   1. Initial program 24.3%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 37.5%

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

      1. pow1/2 N/A

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

      2. unpow-prod-down N/A

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

      3. associate-/l* N/A

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

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

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

   6. Applied egg-rr 53.4%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

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

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

   8. Applied egg-rr 53.4%

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

      1. pow1/2 N/A

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

      2. associate-*r* N/A

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

      3. unpow-prod-down N/A

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

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

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

      5. pow1/2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. pow1/2 N/A

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

      13. sqrt-lowering-sqrt.f64 3.3%

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

   10. Applied egg-rr 3.3%

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

       1. associate-/r/ N/A

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

       2. sqrt-unprod N/A

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

       3. associate-*r* N/A

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

       4. associate-*l/ N/A

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

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

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

   12. Applied egg-rr 37.5%

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

   if 4.39999999999999993e-305 < F < 9.7999999999999998e-11

   1. Initial program 18.1%

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

   3. Taylor expanded in C around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. hypot-define N/A

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

      13. hypot-lowering-hypot.f64 23.5%

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

   5. Simplified 23.5%

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

   if 9.7999999999999998e-11 < F

   1. Initial program 14.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 19.7%

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

   5. Simplified 19.7%

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

4. Final simplification 23.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 4.4 \cdot 10^{-305}:\\ \;\;\;\;\frac{\sqrt{\left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)\right)}}{4 \cdot \left(A \cdot C\right) - B \cdot B}\\ \mathbf{elif}\;F \leq 9.8 \cdot 10^{-11}:\\ \;\;\;\;\left(0 - \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(0 - \sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 36.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := B\_m \cdot B\_m + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B\_m \leq 1.95 \cdot 10^{-178}:\\ \;\;\;\;\frac{\left(\sqrt{t\_0 \cdot \left(2 \cdot F\right)} \cdot 0.25\right) \cdot \sqrt{\frac{2}{C}}}{A}\\ \mathbf{elif}\;B\_m \leq 10^{+71}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(t\_0 \cdot \left(A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B\_m}} \cdot \left(0 - \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (+ (* B_m B_m) (* -4.0 (* A C)))))
   (if (<= B_m 1.95e-178)
     (/ (* (* (sqrt (* t_0 (* 2.0 F))) 0.25) (sqrt (/ 2.0 C))) A)
     (if (<= B_m 1e+71)
       (/
        (sqrt (* (* 2.0 F) (* t_0 (+ A (+ C (hypot B_m (- A C)))))))
        (- (* (* 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 t_0 = (B_m * B_m) + (-4.0 * (A * C));
	double tmp;
	if (B_m <= 1.95e-178) {
		tmp = ((sqrt((t_0 * (2.0 * F))) * 0.25) * sqrt((2.0 / C))) / A;
	} else if (B_m <= 1e+71) {
		tmp = sqrt(((2.0 * F) * (t_0 * (A + (C + hypot(B_m, (A - C))))))) / (((4.0 * A) * C) - (B_m * B_m));
	} 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 t_0 = (B_m * B_m) + (-4.0 * (A * C));
	double tmp;
	if (B_m <= 1.95e-178) {
		tmp = ((Math.sqrt((t_0 * (2.0 * F))) * 0.25) * Math.sqrt((2.0 / C))) / A;
	} else if (B_m <= 1e+71) {
		tmp = Math.sqrt(((2.0 * F) * (t_0 * (A + (C + Math.hypot(B_m, (A - C))))))) / (((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):
	t_0 = (B_m * B_m) + (-4.0 * (A * C))
	tmp = 0
	if B_m <= 1.95e-178:
		tmp = ((math.sqrt((t_0 * (2.0 * F))) * 0.25) * math.sqrt((2.0 / C))) / A
	elif B_m <= 1e+71:
		tmp = math.sqrt(((2.0 * F) * (t_0 * (A + (C + math.hypot(B_m, (A - C))))))) / (((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)
	t_0 = Float64(Float64(B_m * B_m) + Float64(-4.0 * Float64(A * C)))
	tmp = 0.0
	if (B_m <= 1.95e-178)
		tmp = Float64(Float64(Float64(sqrt(Float64(t_0 * Float64(2.0 * F))) * 0.25) * sqrt(Float64(2.0 / C))) / A);
	elseif (B_m <= 1e+71)
		tmp = Float64(sqrt(Float64(Float64(2.0 * F) * Float64(t_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(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)
	t_0 = (B_m * B_m) + (-4.0 * (A * C));
	tmp = 0.0;
	if (B_m <= 1.95e-178)
		tmp = ((sqrt((t_0 * (2.0 * F))) * 0.25) * sqrt((2.0 / C))) / A;
	elseif (B_m <= 1e+71)
		tmp = sqrt(((2.0 * F) * (t_0 * (A + (C + hypot(B_m, (A - C))))))) / (((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_] := Block[{t$95$0 = N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 1.95e-178], N[(N[(N[(N[Sqrt[N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.25), $MachinePrecision] * N[Sqrt[N[(2.0 / C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / A), $MachinePrecision], If[LessEqual[B$95$m, 1e+71], N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(t$95$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[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision] * N[(0.0 - N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.95000000000000013e-178

   1. Initial program 16.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 23.8%

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

      1. pow1/2 N/A

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

      2. unpow-prod-down N/A

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

      3. associate-/l* N/A

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

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

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

   6. Applied egg-rr 31.9%

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

   7. Taylor expanded in A around -inf

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

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

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

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

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

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

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

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

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

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

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

      6. /-lowering-/.f64 13.6%

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

   9. Simplified 13.6%

      \[\leadsto \sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)} \cdot \color{blue}{\left(0.25 \cdot \left(\frac{\sqrt{2}}{A} \cdot \sqrt{\frac{1}{C}}\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

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

       2. associate-*l/ N/A

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

       3. associate-*r/ N/A

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

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

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

   11. Applied egg-rr 13.7%

       \[\leadsto \color{blue}{\frac{\left(\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)} \cdot 0.25\right) \cdot \sqrt{\frac{2}{C}}}{A}} \]

   if 1.95000000000000013e-178 < B < 1e71

   1. Initial program 26.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. 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 34.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing
   5. 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 33.3%

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

   if 1e71 < B

   1. Initial program 9.0%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

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

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

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

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

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

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

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

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

      6. sqrt-lowering-sqrt.f64 52.0%

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

   5. Simplified 52.0%

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

4. Final simplification 24.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.95 \cdot 10^{-178}:\\ \;\;\;\;\frac{\left(\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)} \cdot 0.25\right) \cdot \sqrt{\frac{2}{C}}}{A}\\ \mathbf{elif}\;B \leq 10^{+71}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(0 - \sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 34.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(B\_m, A - C\right)\\ t_1 := B\_m \cdot B\_m - \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;B\_m \leq 1.45 \cdot 10^{-76}:\\ \;\;\;\;\frac{-1}{\frac{t\_1}{\sqrt{\left(-4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(A + \left(C + t\_0\right)\right) \cdot \left(2 \cdot F\right)\right)}}}\\ \mathbf{elif}\;B\_m \leq 5.2 \cdot 10^{+70}:\\ \;\;\;\;\frac{\sqrt{\left(\left(2 \cdot F\right) \cdot t\_1\right) \cdot \left(\left(A + C\right) + t\_0\right)}}{0 - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B\_m}} \cdot \left(0 - \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (hypot B_m (- A C))) (t_1 (- (* B_m B_m) (* (* 4.0 A) C))))
   (if (<= B_m 1.45e-76)
     (/ -1.0 (/ t_1 (sqrt (* (* -4.0 (* A C)) (* (+ A (+ C t_0)) (* 2.0 F))))))
     (if (<= B_m 5.2e+70)
       (/ (sqrt (* (* (* 2.0 F) t_1) (+ (+ A C) t_0))) (- 0.0 (* 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 t_0 = hypot(B_m, (A - C));
	double t_1 = (B_m * B_m) - ((4.0 * A) * C);
	double tmp;
	if (B_m <= 1.45e-76) {
		tmp = -1.0 / (t_1 / sqrt(((-4.0 * (A * C)) * ((A + (C + t_0)) * (2.0 * F)))));
	} else if (B_m <= 5.2e+70) {
		tmp = sqrt((((2.0 * F) * t_1) * ((A + C) + t_0))) / (0.0 - (B_m * B_m));
	} 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 t_0 = Math.hypot(B_m, (A - C));
	double t_1 = (B_m * B_m) - ((4.0 * A) * C);
	double tmp;
	if (B_m <= 1.45e-76) {
		tmp = -1.0 / (t_1 / Math.sqrt(((-4.0 * (A * C)) * ((A + (C + t_0)) * (2.0 * F)))));
	} else if (B_m <= 5.2e+70) {
		tmp = Math.sqrt((((2.0 * F) * t_1) * ((A + C) + t_0))) / (0.0 - (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):
	t_0 = math.hypot(B_m, (A - C))
	t_1 = (B_m * B_m) - ((4.0 * A) * C)
	tmp = 0
	if B_m <= 1.45e-76:
		tmp = -1.0 / (t_1 / math.sqrt(((-4.0 * (A * C)) * ((A + (C + t_0)) * (2.0 * F)))))
	elif B_m <= 5.2e+70:
		tmp = math.sqrt((((2.0 * F) * t_1) * ((A + C) + t_0))) / (0.0 - (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)
	t_0 = hypot(B_m, Float64(A - C))
	t_1 = Float64(Float64(B_m * B_m) - Float64(Float64(4.0 * A) * C))
	tmp = 0.0
	if (B_m <= 1.45e-76)
		tmp = Float64(-1.0 / Float64(t_1 / sqrt(Float64(Float64(-4.0 * Float64(A * C)) * Float64(Float64(A + Float64(C + t_0)) * Float64(2.0 * F))))));
	elseif (B_m <= 5.2e+70)
		tmp = Float64(sqrt(Float64(Float64(Float64(2.0 * F) * t_1) * Float64(Float64(A + C) + t_0))) / Float64(0.0 - 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)
	t_0 = hypot(B_m, (A - C));
	t_1 = (B_m * B_m) - ((4.0 * A) * C);
	tmp = 0.0;
	if (B_m <= 1.45e-76)
		tmp = -1.0 / (t_1 / sqrt(((-4.0 * (A * C)) * ((A + (C + t_0)) * (2.0 * F)))));
	elseif (B_m <= 5.2e+70)
		tmp = sqrt((((2.0 * F) * t_1) * ((A + C) + t_0))) / (0.0 - (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_] := 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[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 1.45e-76], N[(-1.0 / N[(t$95$1 / N[Sqrt[N[(N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] * N[(N[(A + N[(C + t$95$0), $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 5.2e+70], N[(N[Sqrt[N[(N[(N[(2.0 * F), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(0.0 - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision] * N[(0.0 - N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.4500000000000001e-76

   1. Initial program 18.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 25.9%

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

   6. Applied egg-rr 26.0%

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

   7. Taylor expanded in B around 0

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

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

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

      2. *-lowering-*.f64 16.4%

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

   9. Simplified 16.4%

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

   if 1.4500000000000001e-76 < B < 5.2000000000000001e70

   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 29.1%

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

   5. Taylor expanded in A around 0

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 28.8%

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

   7. Simplified 28.8%

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

   if 5.2000000000000001e70 < B

   1. Initial program 9.0%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

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

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

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

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

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

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

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

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

      6. sqrt-lowering-sqrt.f64 52.0%

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

   5. Simplified 52.0%

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

4. Final simplification 24.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.45 \cdot 10^{-76}:\\ \;\;\;\;\frac{-1}{\frac{B \cdot B - \left(4 \cdot A\right) \cdot C}{\sqrt{\left(-4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}}\\ \mathbf{elif}\;B \leq 5.2 \cdot 10^{+70}:\\ \;\;\;\;\frac{\sqrt{\left(\left(2 \cdot F\right) \cdot \left(B \cdot B - \left(4 \cdot A\right) \cdot C\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{0 - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(0 - \sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 34.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \sqrt{\left(\left(2 \cdot F\right) \cdot \left(B\_m \cdot B\_m - \left(4 \cdot A\right) \cdot C\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B\_m, A - C\right)\right)}\\ \mathbf{if}\;B\_m \leq 6.2 \cdot 10^{-76}:\\ \;\;\;\;\frac{t\_0}{4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B\_m \leq 4.1 \cdot 10^{+71}:\\ \;\;\;\;\frac{t\_0}{0 - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B\_m}} \cdot \left(0 - \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0
         (sqrt
          (*
           (* (* 2.0 F) (- (* B_m B_m) (* (* 4.0 A) C)))
           (+ (+ A C) (hypot B_m (- A C)))))))
   (if (<= B_m 6.2e-76)
     (/ t_0 (* 4.0 (* A C)))
     (if (<= B_m 4.1e+71)
       (/ t_0 (- 0.0 (* 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 t_0 = sqrt((((2.0 * F) * ((B_m * B_m) - ((4.0 * A) * C))) * ((A + C) + hypot(B_m, (A - C)))));
	double tmp;
	if (B_m <= 6.2e-76) {
		tmp = t_0 / (4.0 * (A * C));
	} else if (B_m <= 4.1e+71) {
		tmp = t_0 / (0.0 - (B_m * B_m));
	} 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 t_0 = Math.sqrt((((2.0 * F) * ((B_m * B_m) - ((4.0 * A) * C))) * ((A + C) + Math.hypot(B_m, (A - C)))));
	double tmp;
	if (B_m <= 6.2e-76) {
		tmp = t_0 / (4.0 * (A * C));
	} else if (B_m <= 4.1e+71) {
		tmp = t_0 / (0.0 - (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):
	t_0 = math.sqrt((((2.0 * F) * ((B_m * B_m) - ((4.0 * A) * C))) * ((A + C) + math.hypot(B_m, (A - C)))))
	tmp = 0
	if B_m <= 6.2e-76:
		tmp = t_0 / (4.0 * (A * C))
	elif B_m <= 4.1e+71:
		tmp = t_0 / (0.0 - (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)
	t_0 = sqrt(Float64(Float64(Float64(2.0 * F) * Float64(Float64(B_m * B_m) - Float64(Float64(4.0 * A) * C))) * Float64(Float64(A + C) + hypot(B_m, Float64(A - C)))))
	tmp = 0.0
	if (B_m <= 6.2e-76)
		tmp = Float64(t_0 / Float64(4.0 * Float64(A * C)));
	elseif (B_m <= 4.1e+71)
		tmp = Float64(t_0 / Float64(0.0 - 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)
	t_0 = sqrt((((2.0 * F) * ((B_m * B_m) - ((4.0 * A) * C))) * ((A + C) + hypot(B_m, (A - C)))));
	tmp = 0.0;
	if (B_m <= 6.2e-76)
		tmp = t_0 / (4.0 * (A * C));
	elseif (B_m <= 4.1e+71)
		tmp = t_0 / (0.0 - (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_] := Block[{t$95$0 = N[Sqrt[N[(N[(N[(2.0 * F), $MachinePrecision] * N[(N[(B$95$m * B$95$m), $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[B$95$m, 6.2e-76], N[(t$95$0 / N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 4.1e+71], N[(t$95$0 / N[(0.0 - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision] * N[(0.0 - N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 6.19999999999999939e-76

   1. Initial program 18.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 25.9%

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

   5. Taylor expanded in A around inf

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

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

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

      2. *-lowering-*.f64 15.5%

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

   7. Simplified 15.5%

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

   if 6.19999999999999939e-76 < B < 4.1000000000000002e71

   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 29.1%

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

   5. Taylor expanded in A around 0

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 28.8%

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

   7. Simplified 28.8%

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

   if 4.1000000000000002e71 < B

   1. Initial program 9.0%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

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

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

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

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

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

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

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

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

      6. sqrt-lowering-sqrt.f64 52.0%

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

   5. Simplified 52.0%

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

4. Final simplification 24.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 6.2 \cdot 10^{-76}:\\ \;\;\;\;\frac{\sqrt{\left(\left(2 \cdot F\right) \cdot \left(B \cdot B - \left(4 \cdot A\right) \cdot C\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 4.1 \cdot 10^{+71}:\\ \;\;\;\;\frac{\sqrt{\left(\left(2 \cdot F\right) \cdot \left(B \cdot B - \left(4 \cdot A\right) \cdot C\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{0 - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(0 - \sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 34.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \left(A + C\right) + \mathsf{hypot}\left(B\_m, A - C\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;B\_m \leq 1.95 \cdot 10^{-76}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot t\_0\right)}}{t\_1 - B\_m \cdot B\_m}\\ \mathbf{elif}\;B\_m \leq 2.9 \cdot 10^{+70}:\\ \;\;\;\;\frac{\sqrt{\left(\left(2 \cdot F\right) \cdot \left(B\_m \cdot B\_m - t\_1\right)\right) \cdot t\_0}}{0 - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B\_m}} \cdot \left(0 - \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (+ (+ A C) (hypot B_m (- A C)))) (t_1 (* (* 4.0 A) C)))
   (if (<= B_m 1.95e-76)
     (/ (sqrt (* (* -4.0 (* A C)) (* (* 2.0 F) t_0))) (- t_1 (* B_m B_m)))
     (if (<= B_m 2.9e+70)
       (/ (sqrt (* (* (* 2.0 F) (- (* B_m B_m) t_1)) t_0)) (- 0.0 (* 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 t_0 = (A + C) + hypot(B_m, (A - C));
	double t_1 = (4.0 * A) * C;
	double tmp;
	if (B_m <= 1.95e-76) {
		tmp = sqrt(((-4.0 * (A * C)) * ((2.0 * F) * t_0))) / (t_1 - (B_m * B_m));
	} else if (B_m <= 2.9e+70) {
		tmp = sqrt((((2.0 * F) * ((B_m * B_m) - t_1)) * t_0)) / (0.0 - (B_m * B_m));
	} 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 t_0 = (A + C) + Math.hypot(B_m, (A - C));
	double t_1 = (4.0 * A) * C;
	double tmp;
	if (B_m <= 1.95e-76) {
		tmp = Math.sqrt(((-4.0 * (A * C)) * ((2.0 * F) * t_0))) / (t_1 - (B_m * B_m));
	} else if (B_m <= 2.9e+70) {
		tmp = Math.sqrt((((2.0 * F) * ((B_m * B_m) - t_1)) * t_0)) / (0.0 - (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):
	t_0 = (A + C) + math.hypot(B_m, (A - C))
	t_1 = (4.0 * A) * C
	tmp = 0
	if B_m <= 1.95e-76:
		tmp = math.sqrt(((-4.0 * (A * C)) * ((2.0 * F) * t_0))) / (t_1 - (B_m * B_m))
	elif B_m <= 2.9e+70:
		tmp = math.sqrt((((2.0 * F) * ((B_m * B_m) - t_1)) * t_0)) / (0.0 - (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)
	t_0 = Float64(Float64(A + C) + hypot(B_m, Float64(A - C)))
	t_1 = Float64(Float64(4.0 * A) * C)
	tmp = 0.0
	if (B_m <= 1.95e-76)
		tmp = Float64(sqrt(Float64(Float64(-4.0 * Float64(A * C)) * Float64(Float64(2.0 * F) * t_0))) / Float64(t_1 - Float64(B_m * B_m)));
	elseif (B_m <= 2.9e+70)
		tmp = Float64(sqrt(Float64(Float64(Float64(2.0 * F) * Float64(Float64(B_m * B_m) - t_1)) * t_0)) / Float64(0.0 - 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)
	t_0 = (A + C) + hypot(B_m, (A - C));
	t_1 = (4.0 * A) * C;
	tmp = 0.0;
	if (B_m <= 1.95e-76)
		tmp = sqrt(((-4.0 * (A * C)) * ((2.0 * F) * t_0))) / (t_1 - (B_m * B_m));
	elseif (B_m <= 2.9e+70)
		tmp = sqrt((((2.0 * F) * ((B_m * B_m) - t_1)) * t_0)) / (0.0 - (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_] := Block[{t$95$0 = N[(N[(A + C), $MachinePrecision] + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, If[LessEqual[B$95$m, 1.95e-76], N[(N[Sqrt[N[(N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * F), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$1 - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 2.9e+70], N[(N[Sqrt[N[(N[(N[(2.0 * F), $MachinePrecision] * N[(N[(B$95$m * B$95$m), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] / N[(0.0 - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision] * N[(0.0 - N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.95000000000000013e-76

   1. Initial program 18.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 25.9%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   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 33.2%

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

   7. Taylor expanded in B around 0

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

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

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

      2. *-lowering-*.f64 21.4%

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

   9. Simplified 21.4%

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

       1. sqrt-unprod N/A

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

       2. associate-*l* N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. Applied egg-rr 15.3%

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

   if 1.95000000000000013e-76 < B < 2.8999999999999998e70

   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 29.1%

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

   5. Taylor expanded in A around 0

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 28.8%

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

   7. Simplified 28.8%

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

   if 2.8999999999999998e70 < B

   1. Initial program 9.0%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

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

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

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

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

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

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

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

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

      6. sqrt-lowering-sqrt.f64 52.0%

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

   5. Simplified 52.0%

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

4. Final simplification 23.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.95 \cdot 10^{-76}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;B \leq 2.9 \cdot 10^{+70}:\\ \;\;\;\;\frac{\sqrt{\left(\left(2 \cdot F\right) \cdot \left(B \cdot B - \left(4 \cdot A\right) \cdot C\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{0 - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(0 - \sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 38.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 1.95 \cdot 10^{+70}:\\ \;\;\;\;\frac{\sqrt{\left(C + \left(A + \mathsf{hypot}\left(B\_m, A - C\right)\right)\right) \cdot \left(\left(B\_m \cdot B\_m + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)\right)}}{4 \cdot \left(A \cdot C\right) - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B\_m}} \cdot \left(0 - \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 1.95e+70)
   (/
    (sqrt
     (*
      (+ C (+ A (hypot B_m (- A C))))
      (* (+ (* B_m B_m) (* -4.0 (* A C))) (* 2.0 F))))
    (- (* 4.0 (* A C)) (* B_m B_m)))
   (* (sqrt (/ F B_m)) (- 0.0 (sqrt 2.0)))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.95e+70) {
		tmp = sqrt(((C + (A + hypot(B_m, (A - C)))) * (((B_m * B_m) + (-4.0 * (A * C))) * (2.0 * F)))) / ((4.0 * (A * C)) - (B_m * B_m));
	} else {
		tmp = sqrt((F / B_m)) * (0.0 - sqrt(2.0));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.95e+70) {
		tmp = Math.sqrt(((C + (A + Math.hypot(B_m, (A - C)))) * (((B_m * B_m) + (-4.0 * (A * C))) * (2.0 * F)))) / ((4.0 * (A * C)) - (B_m * B_m));
	} else {
		tmp = Math.sqrt((F / B_m)) * (0.0 - Math.sqrt(2.0));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 1.95e+70:
		tmp = math.sqrt(((C + (A + math.hypot(B_m, (A - C)))) * (((B_m * B_m) + (-4.0 * (A * C))) * (2.0 * F)))) / ((4.0 * (A * C)) - (B_m * B_m))
	else:
		tmp = math.sqrt((F / B_m)) * (0.0 - math.sqrt(2.0))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 1.95e+70)
		tmp = Float64(sqrt(Float64(Float64(C + Float64(A + hypot(B_m, Float64(A - C)))) * Float64(Float64(Float64(B_m * B_m) + Float64(-4.0 * Float64(A * C))) * Float64(2.0 * F)))) / Float64(Float64(4.0 * Float64(A * C)) - Float64(B_m * B_m)));
	else
		tmp = Float64(sqrt(Float64(F / B_m)) * Float64(0.0 - sqrt(2.0)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 1.95e+70)
		tmp = sqrt(((C + (A + hypot(B_m, (A - C)))) * (((B_m * B_m) + (-4.0 * (A * C))) * (2.0 * F)))) / ((4.0 * (A * C)) - (B_m * B_m));
	else
		tmp = sqrt((F / B_m)) * (0.0 - sqrt(2.0));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 1.95e+70], N[(N[Sqrt[N[(N[(C + N[(A + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(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 < 1.94999999999999987e70

   1. Initial program 19.3%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 26.4%

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

      1. pow1/2 N/A

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

      2. unpow-prod-down N/A

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

      3. associate-/l* N/A

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

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

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

   6. Applied egg-rr 34.9%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

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

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

   8. Applied egg-rr 34.9%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)}}{\frac{A \cdot \left(4 \cdot C\right) - B \cdot B}{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}}}} \]
   9. Step-by-step derivation

      1. pow1/2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)\right)}^{\frac{1}{2}}\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{\_.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(4, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot 2\right) \cdot F\right)}^{\frac{1}{2}}\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(4, C\right)\right)}, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right) \]

      3. unpow-prod-down N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot 2\right)}^{\frac{1}{2}} \cdot {F}^{\frac{1}{2}}\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{\_.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(4, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({\left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot 2\right)}^{\frac{1}{2}}\right), \left({F}^{\frac{1}{2}}\right)\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{\_.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(4, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right) \]

      5. pow1/2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot 2}\right), \left({F}^{\frac{1}{2}}\right)\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(4, C\right)\right)}, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot 2\right)\right), \left({F}^{\frac{1}{2}}\right)\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(4, C\right)\right)}, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right), 2\right)\right), \left({F}^{\frac{1}{2}}\right)\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{A}, \mathsf{*.f64}\left(4, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(B \cdot B\right), \left(-4 \cdot \left(A \cdot C\right)\right)\right), 2\right)\right), \left({F}^{\frac{1}{2}}\right)\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(4, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \left(-4 \cdot \left(A \cdot C\right)\right)\right), 2\right)\right), \left({F}^{\frac{1}{2}}\right)\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(4, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \left(A \cdot C\right)\right)\right), 2\right)\right), \left({F}^{\frac{1}{2}}\right)\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(4, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right)\right), 2\right)\right), \left({F}^{\frac{1}{2}}\right)\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(4, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right) \]

      12. pow1/2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right)\right), 2\right)\right), \left(\sqrt{F}\right)\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(4, C\right)\right), \color{blue}{\mathsf{*.f64}\left(B, B\right)}\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right) \]

      13. sqrt-lowering-sqrt.f64 32.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right)\right), 2\right)\right), \mathsf{sqrt.f64}\left(F\right)\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(4, C\right)\right), \color{blue}{\mathsf{*.f64}\left(B, B\right)}\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right) \]

   10. Applied egg-rr 32.3%

       \[\leadsto \frac{\color{blue}{\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot 2} \cdot \sqrt{F}}}{\frac{A \cdot \left(4 \cdot C\right) - B \cdot B}{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}}} \]
   11. Step-by-step derivation

       1. associate-/r/ N/A

          \[\leadsto \frac{\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot 2} \cdot \sqrt{F}}{A \cdot \left(4 \cdot C\right) - B \cdot B} \cdot \color{blue}{\sqrt{A + \left(C + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)}} \]

       2. sqrt-unprod N/A

          \[\leadsto \frac{\sqrt{\left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot 2\right) \cdot F}}{A \cdot \left(4 \cdot C\right) - B \cdot B} \cdot \sqrt{\color{blue}{A} + \left(C + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)} \]

       3. associate-*r* N/A

          \[\leadsto \frac{\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)}}{A \cdot \left(4 \cdot C\right) - B \cdot B} \cdot \sqrt{A + \left(C + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)} \]

       4. associate-*l/ N/A

          \[\leadsto \frac{\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)} \cdot \sqrt{A + \left(C + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)}}{\color{blue}{A \cdot \left(4 \cdot C\right) - B \cdot B}} \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)} \cdot \sqrt{A + \left(C + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)}\right), \color{blue}{\left(A \cdot \left(4 \cdot C\right) - B \cdot B\right)}\right) \]

   12. Applied egg-rr 27.5%

       \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(C + \left(\mathsf{hypot}\left(B, A - C\right) + A\right)\right)}}{\left(A \cdot C\right) \cdot 4 - B \cdot B}} \]

   if 1.94999999999999987e70 < B

   1. Initial program 9.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{F}{B}}\right), \left(\sqrt{2}\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{F}{B}\right)\right), \left(\sqrt{2}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \left(\sqrt{2}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 52.0%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   5. Simplified 52.0%

      \[\leadsto \color{blue}{-\sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.95 \cdot 10^{+70}:\\ \;\;\;\;\frac{\sqrt{\left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)\right)}}{4 \cdot \left(A \cdot C\right) - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(0 - \sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 33.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 2.1 \cdot 10^{-76}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B\_m, A - C\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B\_m}} \cdot \left(0 - \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 2.1e-76)
   (/
    (sqrt (* (* -4.0 (* A C)) (* (* 2.0 F) (+ (+ A C) (hypot B_m (- A C))))))
    (- (* (* 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 <= 2.1e-76) {
		tmp = sqrt(((-4.0 * (A * C)) * ((2.0 * F) * ((A + C) + hypot(B_m, (A - C)))))) / (((4.0 * A) * C) - (B_m * B_m));
	} else {
		tmp = sqrt((F / B_m)) * (0.0 - sqrt(2.0));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 2.1e-76) {
		tmp = Math.sqrt(((-4.0 * (A * C)) * ((2.0 * F) * ((A + C) + Math.hypot(B_m, (A - C)))))) / (((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 <= 2.1e-76:
		tmp = math.sqrt(((-4.0 * (A * C)) * ((2.0 * F) * ((A + C) + math.hypot(B_m, (A - C)))))) / (((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 <= 2.1e-76)
		tmp = Float64(sqrt(Float64(Float64(-4.0 * Float64(A * C)) * Float64(Float64(2.0 * F) * Float64(Float64(A + C) + hypot(B_m, Float64(A - C)))))) / 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 <= 2.1e-76)
		tmp = sqrt(((-4.0 * (A * C)) * ((2.0 * F) * ((A + C) + hypot(B_m, (A - C)))))) / (((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, 2.1e-76], N[(N[Sqrt[N[(N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * F), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 < 2.09999999999999992e-76

   1. Initial program 18.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 25.9%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   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 33.2%

      \[\leadsto \frac{\color{blue}{\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot F} \cdot \sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   7. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(-4 \cdot \left(A \cdot C\right)\right)}, F\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(A, \mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \left(A \cdot C\right)\right), F\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(A, \mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 21.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right), F\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(A, \mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Simplified 21.4%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-4 \cdot \left(A \cdot C\right)\right)} \cdot F} \cdot \sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
   10. Step-by-step derivation

       1. sqrt-unprod N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(\left(-4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)}\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)}, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(-4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(2 \cdot \left(A + \left(C + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(-4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(F \cdot 2\right) \cdot \left(A + \left(C + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(-4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(A + \left(C + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       5. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(-4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(A + \left(C + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)}, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(-4 \cdot \left(A \cdot C\right)\right), \left(\left(2 \cdot F\right) \cdot \left(A + \left(C + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \left(A \cdot C\right)\right), \left(\left(2 \cdot F\right) \cdot \left(A + \left(C + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right), \left(\left(2 \cdot F\right) \cdot \left(A + \left(C + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(\left(2 \cdot F\right), \left(A + \left(C + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, F\right), \left(A + \left(C + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   11. Applied egg-rr 15.3%

       \[\leadsto \frac{\color{blue}{\sqrt{\left(-4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if 2.09999999999999992e-76 < B

   1. Initial program 15.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in 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 39.9%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   5. Simplified 39.9%

      \[\leadsto \color{blue}{-\sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 23.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.1 \cdot 10^{-76}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(0 - \sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 32.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 1.15 \cdot 10^{-108}:\\ \;\;\;\;\frac{\left(\sqrt{\left(B\_m \cdot B\_m + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)} \cdot 0.25\right) \cdot \sqrt{\frac{2}{C}}}{A}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B\_m}} \cdot \left(0 - \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 1.15e-108)
   (/
    (*
     (* (sqrt (* (+ (* B_m B_m) (* -4.0 (* A C))) (* 2.0 F))) 0.25)
     (sqrt (/ 2.0 C)))
    A)
   (* (sqrt (/ F B_m)) (- 0.0 (sqrt 2.0)))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.15e-108) {
		tmp = ((sqrt((((B_m * B_m) + (-4.0 * (A * C))) * (2.0 * F))) * 0.25) * sqrt((2.0 / C))) / A;
	} else {
		tmp = sqrt((F / B_m)) * (0.0 - sqrt(2.0));
	}
	return tmp;
}

Fortran

B_m = abs(b)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b_m <= 1.15d-108) then
        tmp = ((sqrt((((b_m * b_m) + ((-4.0d0) * (a * c))) * (2.0d0 * f))) * 0.25d0) * sqrt((2.0d0 / c))) / a
    else
        tmp = sqrt((f / b_m)) * (0.0d0 - sqrt(2.0d0))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.15e-108) {
		tmp = ((Math.sqrt((((B_m * B_m) + (-4.0 * (A * C))) * (2.0 * F))) * 0.25) * Math.sqrt((2.0 / C))) / A;
	} else {
		tmp = Math.sqrt((F / B_m)) * (0.0 - Math.sqrt(2.0));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 1.15e-108:
		tmp = ((math.sqrt((((B_m * B_m) + (-4.0 * (A * C))) * (2.0 * F))) * 0.25) * math.sqrt((2.0 / C))) / A
	else:
		tmp = math.sqrt((F / B_m)) * (0.0 - math.sqrt(2.0))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 1.15e-108)
		tmp = Float64(Float64(Float64(sqrt(Float64(Float64(Float64(B_m * B_m) + Float64(-4.0 * Float64(A * C))) * Float64(2.0 * F))) * 0.25) * sqrt(Float64(2.0 / C))) / A);
	else
		tmp = Float64(sqrt(Float64(F / B_m)) * Float64(0.0 - sqrt(2.0)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 1.15e-108)
		tmp = ((sqrt((((B_m * B_m) + (-4.0 * (A * C))) * (2.0 * F))) * 0.25) * sqrt((2.0 / C))) / A;
	else
		tmp = sqrt((F / B_m)) * (0.0 - sqrt(2.0));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 1.15e-108], N[(N[(N[(N[Sqrt[N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.25), $MachinePrecision] * N[Sqrt[N[(2.0 / C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / A), $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 < 1.14999999999999998e-108

   1. Initial program 17.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 25.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. pow1/2 N/A

         \[\leadsto \frac{{\left(\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)}^{\frac{1}{2}}}{\color{blue}{\left(4 \cdot A\right) \cdot C} - B \cdot B} \]

      2. unpow-prod-down N/A

         \[\leadsto \frac{{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right)}^{\frac{1}{2}} \cdot {\left(\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)}^{\frac{1}{2}}}{\color{blue}{\left(4 \cdot A\right) \cdot C} - B \cdot B} \]

      3. associate-/l* N/A

         \[\leadsto {\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right)}^{\frac{1}{2}} \cdot \color{blue}{\frac{{\left(\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)}^{\frac{1}{2}}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right)}^{\frac{1}{2}}\right), \color{blue}{\left(\frac{{\left(\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)}^{\frac{1}{2}}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\right)}\right) \]

   6. Applied egg-rr 33.1%

      \[\leadsto \color{blue}{\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)} \cdot \frac{{\left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{0.5}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]

   7. Taylor expanded in A around -inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right), \color{blue}{\left(\frac{1}{4} \cdot \left(\frac{\sqrt{2}}{A} \cdot \sqrt{\frac{1}{C}}\right)\right)}\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{4}, \color{blue}{\left(\frac{\sqrt{2}}{A} \cdot \sqrt{\frac{1}{C}}\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(\frac{\sqrt{2}}{A}\right), \color{blue}{\left(\sqrt{\frac{1}{C}}\right)}\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2}\right), A\right), \left(\sqrt{\color{blue}{\frac{1}{C}}}\right)\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), A\right), \left(\sqrt{\frac{\color{blue}{1}}{C}}\right)\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), A\right), \mathsf{sqrt.f64}\left(\left(\frac{1}{C}\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 14.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), A\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, C\right)\right)\right)\right)\right) \]

   9. Simplified 14.3%

      \[\leadsto \sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)} \cdot \color{blue}{\left(0.25 \cdot \left(\frac{\sqrt{2}}{A} \cdot \sqrt{\frac{1}{C}}\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)} \cdot \frac{1}{4}\right) \cdot \color{blue}{\left(\frac{\sqrt{2}}{A} \cdot \sqrt{\frac{1}{C}}\right)} \]

       2. associate-*l/ N/A

          \[\leadsto \left(\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)} \cdot \frac{1}{4}\right) \cdot \frac{\sqrt{2} \cdot \sqrt{\frac{1}{C}}}{\color{blue}{A}} \]

       3. associate-*r/ N/A

          \[\leadsto \frac{\left(\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)} \cdot \frac{1}{4}\right) \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{C}}\right)}{\color{blue}{A}} \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)} \cdot \frac{1}{4}\right) \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{C}}\right)\right), \color{blue}{A}\right) \]

   11. Applied egg-rr 14.4%

       \[\leadsto \color{blue}{\frac{\left(\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)} \cdot 0.25\right) \cdot \sqrt{\frac{2}{C}}}{A}} \]

   if 1.14999999999999998e-108 < B

   1. Initial program 16.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   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 37.9%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   5. Simplified 37.9%

      \[\leadsto \color{blue}{-\sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 22.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.15 \cdot 10^{-108}:\\ \;\;\;\;\frac{\left(\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)} \cdot 0.25\right) \cdot \sqrt{\frac{2}{C}}}{A}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(0 - \sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 32.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 1.8 \cdot 10^{-108}:\\ \;\;\;\;\sqrt{\left(B\_m \cdot B\_m + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)} \cdot \frac{0.25 \cdot \sqrt{\frac{2}{C}}}{A}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B\_m}} \cdot \left(0 - \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 1.8e-108)
   (*
    (sqrt (* (+ (* B_m B_m) (* -4.0 (* A C))) (* 2.0 F)))
    (/ (* 0.25 (sqrt (/ 2.0 C))) A))
   (* (sqrt (/ F B_m)) (- 0.0 (sqrt 2.0)))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.8e-108) {
		tmp = sqrt((((B_m * B_m) + (-4.0 * (A * C))) * (2.0 * F))) * ((0.25 * sqrt((2.0 / C))) / A);
	} else {
		tmp = sqrt((F / B_m)) * (0.0 - sqrt(2.0));
	}
	return tmp;
}

Fortran

B_m = abs(b)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b_m <= 1.8d-108) then
        tmp = sqrt((((b_m * b_m) + ((-4.0d0) * (a * c))) * (2.0d0 * f))) * ((0.25d0 * sqrt((2.0d0 / c))) / a)
    else
        tmp = sqrt((f / b_m)) * (0.0d0 - sqrt(2.0d0))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.8e-108) {
		tmp = Math.sqrt((((B_m * B_m) + (-4.0 * (A * C))) * (2.0 * F))) * ((0.25 * Math.sqrt((2.0 / C))) / A);
	} else {
		tmp = Math.sqrt((F / B_m)) * (0.0 - Math.sqrt(2.0));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 1.8e-108:
		tmp = math.sqrt((((B_m * B_m) + (-4.0 * (A * C))) * (2.0 * F))) * ((0.25 * math.sqrt((2.0 / C))) / A)
	else:
		tmp = math.sqrt((F / B_m)) * (0.0 - math.sqrt(2.0))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 1.8e-108)
		tmp = Float64(sqrt(Float64(Float64(Float64(B_m * B_m) + Float64(-4.0 * Float64(A * C))) * Float64(2.0 * F))) * Float64(Float64(0.25 * sqrt(Float64(2.0 / C))) / A));
	else
		tmp = Float64(sqrt(Float64(F / B_m)) * Float64(0.0 - sqrt(2.0)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 1.8e-108)
		tmp = sqrt((((B_m * B_m) + (-4.0 * (A * C))) * (2.0 * F))) * ((0.25 * sqrt((2.0 / C))) / A);
	else
		tmp = sqrt((F / B_m)) * (0.0 - sqrt(2.0));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 1.8e-108], N[(N[Sqrt[N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(0.25 * N[Sqrt[N[(2.0 / C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / A), $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 < 1.8e-108

   1. Initial program 17.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 25.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. pow1/2 N/A

         \[\leadsto \frac{{\left(\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)}^{\frac{1}{2}}}{\color{blue}{\left(4 \cdot A\right) \cdot C} - B \cdot B} \]

      2. unpow-prod-down N/A

         \[\leadsto \frac{{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right)}^{\frac{1}{2}} \cdot {\left(\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)}^{\frac{1}{2}}}{\color{blue}{\left(4 \cdot A\right) \cdot C} - B \cdot B} \]

      3. associate-/l* N/A

         \[\leadsto {\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right)}^{\frac{1}{2}} \cdot \color{blue}{\frac{{\left(\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)}^{\frac{1}{2}}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right)}^{\frac{1}{2}}\right), \color{blue}{\left(\frac{{\left(\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)}^{\frac{1}{2}}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\right)}\right) \]

   6. Applied egg-rr 33.1%

      \[\leadsto \color{blue}{\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)} \cdot \frac{{\left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{0.5}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]

   7. Taylor expanded in A around -inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right), \color{blue}{\left(\frac{1}{4} \cdot \left(\frac{\sqrt{2}}{A} \cdot \sqrt{\frac{1}{C}}\right)\right)}\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{4}, \color{blue}{\left(\frac{\sqrt{2}}{A} \cdot \sqrt{\frac{1}{C}}\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(\frac{\sqrt{2}}{A}\right), \color{blue}{\left(\sqrt{\frac{1}{C}}\right)}\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2}\right), A\right), \left(\sqrt{\color{blue}{\frac{1}{C}}}\right)\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), A\right), \left(\sqrt{\frac{\color{blue}{1}}{C}}\right)\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), A\right), \mathsf{sqrt.f64}\left(\left(\frac{1}{C}\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 14.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), A\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, C\right)\right)\right)\right)\right) \]

   9. Simplified 14.3%

      \[\leadsto \sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)} \cdot \color{blue}{\left(0.25 \cdot \left(\frac{\sqrt{2}}{A} \cdot \sqrt{\frac{1}{C}}\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\frac{1}{4} \cdot \left(\frac{\sqrt{2}}{A} \cdot \sqrt{\frac{1}{C}}\right)\right) \cdot \color{blue}{\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)}} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{4} \cdot \left(\frac{\sqrt{2}}{A} \cdot \sqrt{\frac{1}{C}}\right)\right), \color{blue}{\left(\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)}\right)}\right) \]

       3. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{4} \cdot \frac{\sqrt{2} \cdot \sqrt{\frac{1}{C}}}{A}\right), \left(\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \color{blue}{\left(2 \cdot F\right)}}\right)\right) \]

       4. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{4} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{C}}\right)}{A}\right), \left(\sqrt{\color{blue}{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)}}\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{4} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{C}}\right)\right), A\right), \left(\sqrt{\color{blue}{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)}}\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(\sqrt{2} \cdot \sqrt{\frac{1}{C}}\right)\right), A\right), \left(\sqrt{\color{blue}{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)} \cdot \left(2 \cdot F\right)}\right)\right) \]

       7. sqrt-div N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(\sqrt{2} \cdot \frac{\sqrt{1}}{\sqrt{C}}\right)\right), A\right), \left(\sqrt{\left(B \cdot B + -4 \cdot \color{blue}{\left(A \cdot C\right)}\right) \cdot \left(2 \cdot F\right)}\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(\sqrt{2} \cdot \frac{1}{\sqrt{C}}\right)\right), A\right), \left(\sqrt{\left(B \cdot B + -4 \cdot \left(\color{blue}{A} \cdot C\right)\right) \cdot \left(2 \cdot F\right)}\right)\right) \]

       9. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(\frac{\sqrt{2}}{\sqrt{C}}\right)\right), A\right), \left(\sqrt{\left(B \cdot B + \color{blue}{-4 \cdot \left(A \cdot C\right)}\right) \cdot \left(2 \cdot F\right)}\right)\right) \]

       10. sqrt-undiv N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(\sqrt{\frac{2}{C}}\right)\right), A\right), \left(\sqrt{\left(B \cdot B + \color{blue}{-4 \cdot \left(A \cdot C\right)}\right) \cdot \left(2 \cdot F\right)}\right)\right) \]

       11. sqrt-lowering-sqrt.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{sqrt.f64}\left(\left(\frac{2}{C}\right)\right)\right), A\right), \left(\sqrt{\left(B \cdot B + \color{blue}{-4 \cdot \left(A \cdot C\right)}\right) \cdot \left(2 \cdot F\right)}\right)\right) \]

       12. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, C\right)\right)\right), A\right), \left(\sqrt{\left(B \cdot B + \color{blue}{-4} \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)}\right)\right) \]

       13. sqrt-lowering-sqrt.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, C\right)\right)\right), A\right), \mathsf{sqrt.f64}\left(\left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)\right)\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, C\right)\right)\right), A\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right), \left(2 \cdot F\right)\right)\right)\right) \]

   11. Applied egg-rr 14.4%

       \[\leadsto \color{blue}{\frac{0.25 \cdot \sqrt{\frac{2}{C}}}{A} \cdot \sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)}} \]

   if 1.8e-108 < B

   1. Initial program 16.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   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 37.9%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   5. Simplified 37.9%

      \[\leadsto \color{blue}{-\sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 22.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.8 \cdot 10^{-108}:\\ \;\;\;\;\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)} \cdot \frac{0.25 \cdot \sqrt{\frac{2}{C}}}{A}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(0 - \sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 29.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 2.8 \cdot 10^{-100}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -16\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B\_m}} \cdot \left(0 - \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 2.8e-100)
   (/ (sqrt (* (* A -16.0) (* F (* C C)))) (- (* (* 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 <= 2.8e-100) {
		tmp = sqrt(((A * -16.0) * (F * (C * C)))) / (((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 <= 2.8d-100) then
        tmp = sqrt(((a * (-16.0d0)) * (f * (c * c)))) / (((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 <= 2.8e-100) {
		tmp = Math.sqrt(((A * -16.0) * (F * (C * C)))) / (((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 <= 2.8e-100:
		tmp = math.sqrt(((A * -16.0) * (F * (C * C)))) / (((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 <= 2.8e-100)
		tmp = Float64(sqrt(Float64(Float64(A * -16.0) * Float64(F * Float64(C * C)))) / 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 <= 2.8e-100)
		tmp = sqrt(((A * -16.0) * (F * (C * C)))) / (((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, 2.8e-100], N[(N[Sqrt[N[(N[(A * -16.0), $MachinePrecision] * N[(F * N[(C * C), $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 < 2.79999999999999995e-100

   1. Initial program 18.3%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 26.3%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   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 13.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, A\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(C, C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 13.2%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-16 \cdot A\right) \cdot \left(\left(C \cdot C\right) \cdot F\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if 2.79999999999999995e-100 < B

   1. Initial program 15.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 38.7%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   5. Simplified 38.7%

      \[\leadsto \color{blue}{-\sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 21.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.8 \cdot 10^{-100}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -16\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(0 - \sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 26.0% accurate, 5.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := F \cdot \left(C \cdot C\right)\\ \mathbf{if}\;B\_m \leq 1.35 \cdot 10^{-99}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -16\right) \cdot t\_0}}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;0 - \sqrt{\frac{2 \cdot \left(F + \frac{C \cdot F}{B\_m}\right) + \frac{t\_0}{B\_m \cdot B\_m}}{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* F (* C C))))
   (if (<= B_m 1.35e-99)
     (/ (sqrt (* (* A -16.0) t_0)) (- (* (* 4.0 A) C) (* B_m B_m)))
     (-
      0.0
      (sqrt (/ (+ (* 2.0 (+ F (/ (* C F) B_m))) (/ t_0 (* B_m B_m))) B_m))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = F * (C * C);
	double tmp;
	if (B_m <= 1.35e-99) {
		tmp = sqrt(((A * -16.0) * t_0)) / (((4.0 * A) * C) - (B_m * B_m));
	} else {
		tmp = 0.0 - sqrt((((2.0 * (F + ((C * F) / B_m))) + (t_0 / (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) :: t_0
    real(8) :: tmp
    t_0 = f * (c * c)
    if (b_m <= 1.35d-99) then
        tmp = sqrt(((a * (-16.0d0)) * t_0)) / (((4.0d0 * a) * c) - (b_m * b_m))
    else
        tmp = 0.0d0 - sqrt((((2.0d0 * (f + ((c * f) / b_m))) + (t_0 / (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 t_0 = F * (C * C);
	double tmp;
	if (B_m <= 1.35e-99) {
		tmp = Math.sqrt(((A * -16.0) * t_0)) / (((4.0 * A) * C) - (B_m * B_m));
	} else {
		tmp = 0.0 - Math.sqrt((((2.0 * (F + ((C * F) / B_m))) + (t_0 / (B_m * B_m))) / B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = F * (C * C)
	tmp = 0
	if B_m <= 1.35e-99:
		tmp = math.sqrt(((A * -16.0) * t_0)) / (((4.0 * A) * C) - (B_m * B_m))
	else:
		tmp = 0.0 - math.sqrt((((2.0 * (F + ((C * F) / B_m))) + (t_0 / (B_m * B_m))) / B_m))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(F * Float64(C * C))
	tmp = 0.0
	if (B_m <= 1.35e-99)
		tmp = Float64(sqrt(Float64(Float64(A * -16.0) * t_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(t_0 / 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)
	t_0 = F * (C * C);
	tmp = 0.0;
	if (B_m <= 1.35e-99)
		tmp = sqrt(((A * -16.0) * t_0)) / (((4.0 * A) * C) - (B_m * B_m));
	else
		tmp = 0.0 - sqrt((((2.0 * (F + ((C * F) / B_m))) + (t_0 / (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_] := Block[{t$95$0 = N[(F * N[(C * C), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 1.35e-99], N[(N[Sqrt[N[(N[(A * -16.0), $MachinePrecision] * t$95$0), $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[(t$95$0 / 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 < 1.35e-99

   1. Initial program 18.3%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 26.3%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   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 13.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, A\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(C, C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 13.2%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-16 \cdot A\right) \cdot \left(\left(C \cdot C\right) \cdot F\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if 1.35e-99 < B

   1. Initial program 15.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 17.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   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 6.6%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(B \cdot \left(B \cdot B\right)\right) \cdot \left(2 \cdot \left(F + F \cdot \frac{C + A}{B}\right) + \frac{2 \cdot F}{B} \cdot \frac{-4 \cdot \left(A \cdot C\right) + 0.5 \cdot \left(\left(A - C\right) \cdot \left(A - C\right)\right)}{B}\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   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 33.3%

       \[\leadsto \color{blue}{-\sqrt{\frac{2 \cdot \left(F + \frac{C \cdot F}{B}\right) + \frac{\left(C \cdot C\right) \cdot F}{B \cdot B}}{B}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 19.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.35 \cdot 10^{-99}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -16\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;0 - \sqrt{\frac{2 \cdot \left(F + \frac{C \cdot F}{B}\right) + \frac{F \cdot \left(C \cdot C\right)}{B \cdot B}}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 25.9% accurate, 5.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 2.35 \cdot 10^{-92}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -16\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;0 - \sqrt{\frac{2 \cdot \left(F + \frac{A \cdot F}{B\_m}\right) + \frac{F \cdot \left(A \cdot A\right)}{B\_m \cdot B\_m}}{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 2.35e-92)
   (/ (sqrt (* (* A -16.0) (* F (* 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 <= 2.35e-92) {
		tmp = sqrt(((A * -16.0) * (F * (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 <= 2.35d-92) then
        tmp = sqrt(((a * (-16.0d0)) * (f * (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 <= 2.35e-92) {
		tmp = Math.sqrt(((A * -16.0) * (F * (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 <= 2.35e-92:
		tmp = math.sqrt(((A * -16.0) * (F * (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 <= 2.35e-92)
		tmp = Float64(sqrt(Float64(Float64(A * -16.0) * Float64(F * 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 <= 2.35e-92)
		tmp = sqrt(((A * -16.0) * (F * (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, 2.35e-92], N[(N[Sqrt[N[(N[(A * -16.0), $MachinePrecision] * N[(F * N[(C * C), $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 < 2.34999999999999996e-92

   1. Initial program 18.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 26.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in 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 13.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, A\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(C, C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 13.1%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-16 \cdot A\right) \cdot \left(\left(C \cdot C\right) \cdot F\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if 2.34999999999999996e-92 < B

   1. Initial program 15.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 18.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   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 6.7%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(B \cdot \left(B \cdot B\right)\right) \cdot \left(2 \cdot \left(F + F \cdot \frac{C + A}{B}\right) + \frac{2 \cdot F}{B} \cdot \frac{-4 \cdot \left(A \cdot C\right) + 0.5 \cdot \left(\left(A - C\right) \cdot \left(A - C\right)\right)}{B}\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   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 30.5%

       \[\leadsto \color{blue}{-\sqrt{\frac{2 \cdot \left(F + \frac{A \cdot F}{B}\right) + \frac{\left(A \cdot A\right) \cdot F}{B \cdot B}}{B}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 18.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.35 \cdot 10^{-92}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -16\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;0 - \sqrt{\frac{2 \cdot \left(F + \frac{A \cdot F}{B}\right) + \frac{F \cdot \left(A \cdot A\right)}{B \cdot B}}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 13.2% accurate, 5.1× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m\\ \mathbf{if}\;B\_m \leq 4.5 \cdot 10^{-56}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -16\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B\_m \cdot \left(B\_m \cdot B\_m\right)\right)\right)}}{t\_0}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (* (* 4.0 A) C) (* B_m B_m))))
   (if (<= B_m 4.5e-56)
     (/ (sqrt (* (* A -16.0) (* F (* C C)))) t_0)
     (/ (sqrt (* 2.0 (* F (* B_m (* B_m B_m))))) t_0))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = ((4.0 * A) * C) - (B_m * B_m);
	double tmp;
	if (B_m <= 4.5e-56) {
		tmp = sqrt(((A * -16.0) * (F * (C * C)))) / t_0;
	} else {
		tmp = sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0;
	}
	return tmp;
}

Fortran

B_m = abs(b)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((4.0d0 * a) * c) - (b_m * b_m)
    if (b_m <= 4.5d-56) then
        tmp = sqrt(((a * (-16.0d0)) * (f * (c * c)))) / t_0
    else
        tmp = sqrt((2.0d0 * (f * (b_m * (b_m * b_m))))) / t_0
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double t_0 = ((4.0 * A) * C) - (B_m * B_m);
	double tmp;
	if (B_m <= 4.5e-56) {
		tmp = Math.sqrt(((A * -16.0) * (F * (C * C)))) / t_0;
	} else {
		tmp = Math.sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = ((4.0 * A) * C) - (B_m * B_m)
	tmp = 0
	if B_m <= 4.5e-56:
		tmp = math.sqrt(((A * -16.0) * (F * (C * C)))) / t_0
	else:
		tmp = math.sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m))
	tmp = 0.0
	if (B_m <= 4.5e-56)
		tmp = Float64(sqrt(Float64(Float64(A * -16.0) * Float64(F * Float64(C * C)))) / t_0);
	else
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(B_m * Float64(B_m * B_m))))) / t_0);
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = ((4.0 * A) * C) - (B_m * B_m);
	tmp = 0.0;
	if (B_m <= 4.5e-56)
		tmp = sqrt(((A * -16.0) * (F * (C * C)))) / t_0;
	else
		tmp = sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 4.5e-56], N[(N[Sqrt[N[(N[(A * -16.0), $MachinePrecision] * N[(F * N[(C * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[Sqrt[N[(2.0 * N[(F * N[(B$95$m * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 4.5000000000000001e-56

   1. Initial program 18.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 25.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   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 12.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, A\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(C, C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 12.7%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-16 \cdot A\right) \cdot \left(\left(C \cdot C\right) \cdot F\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if 4.5000000000000001e-56 < B

   1. Initial program 15.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 18.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in B around 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 7.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 7.7%

      \[\leadsto \frac{\sqrt{\color{blue}{2 \cdot \left(\left(B \cdot \left(B \cdot B\right)\right) \cdot F\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
3. Recombined 2 regimes into one program.

4. Final simplification 11.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 4.5 \cdot 10^{-56}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -16\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot \left(B \cdot B\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 9.7% accurate, 5.1× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m\\ \mathbf{if}\;A \leq 2.65 \cdot 10^{-48}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B\_m \cdot \left(B\_m \cdot B\_m\right)\right)\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{4 \cdot \left(A \cdot \left(F \cdot \left(B\_m \cdot B\_m\right)\right)\right)}}{t\_0}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (* (* 4.0 A) C) (* B_m B_m))))
   (if (<= A 2.65e-48)
     (/ (sqrt (* 2.0 (* F (* B_m (* B_m B_m))))) t_0)
     (/ (sqrt (* 4.0 (* A (* F (* B_m B_m))))) t_0))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = ((4.0 * A) * C) - (B_m * B_m);
	double tmp;
	if (A <= 2.65e-48) {
		tmp = sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0;
	} else {
		tmp = sqrt((4.0 * (A * (F * (B_m * B_m))))) / t_0;
	}
	return tmp;
}

Fortran

B_m = abs(b)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((4.0d0 * a) * c) - (b_m * b_m)
    if (a <= 2.65d-48) then
        tmp = sqrt((2.0d0 * (f * (b_m * (b_m * b_m))))) / t_0
    else
        tmp = sqrt((4.0d0 * (a * (f * (b_m * b_m))))) / t_0
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double t_0 = ((4.0 * A) * C) - (B_m * B_m);
	double tmp;
	if (A <= 2.65e-48) {
		tmp = Math.sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0;
	} else {
		tmp = Math.sqrt((4.0 * (A * (F * (B_m * B_m))))) / t_0;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = ((4.0 * A) * C) - (B_m * B_m)
	tmp = 0
	if A <= 2.65e-48:
		tmp = math.sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0
	else:
		tmp = math.sqrt((4.0 * (A * (F * (B_m * B_m))))) / t_0
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m))
	tmp = 0.0
	if (A <= 2.65e-48)
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(B_m * Float64(B_m * B_m))))) / t_0);
	else
		tmp = Float64(sqrt(Float64(4.0 * Float64(A * Float64(F * Float64(B_m * B_m))))) / t_0);
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = ((4.0 * A) * C) - (B_m * B_m);
	tmp = 0.0;
	if (A <= 2.65e-48)
		tmp = sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0;
	else
		tmp = sqrt((4.0 * (A * (F * (B_m * B_m))))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[A, 2.65e-48], N[(N[Sqrt[N[(2.0 * N[(F * N[(B$95$m * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[Sqrt[N[(4.0 * N[(A * N[(F * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if A < 2.65e-48

   1. Initial program 18.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 21.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   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 4.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 4.5%

      \[\leadsto \frac{\sqrt{\color{blue}{2 \cdot \left(\left(B \cdot \left(B \cdot B\right)\right) \cdot F\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if 2.65e-48 < A

   1. Initial program 15.3%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 28.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right) + {B}^{2} \cdot \left(2 \cdot \left(F \cdot \left(-2 \cdot \frac{A \cdot C}{A - C} + 2 \cdot A\right)\right) + 2 \cdot \left({B}^{2} \cdot \left(F \cdot \left(\frac{1}{2} \cdot \frac{A \cdot C}{{\left(A - C\right)}^{3}} + \frac{1}{2} \cdot \frac{1}{A - C}\right)\right)\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 8.7%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right) + \left(B \cdot B\right) \cdot \left(2 \cdot \left(F \cdot \left(\frac{-2 \cdot \left(A \cdot C\right)}{A - C} + 2 \cdot A\right) + \left(\left(B \cdot B\right) \cdot F\right) \cdot \left(\frac{0.5 \cdot \left(A \cdot C\right)}{\left(A - C\right) \cdot \left(\left(A - C\right) \cdot \left(A - C\right)\right)} + \frac{0.5}{A - C}\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   8. Taylor expanded in C around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left({B}^{2} \cdot \left(\frac{1}{2} \cdot \frac{{B}^{2} \cdot F}{A} + 2 \cdot \left(A \cdot F\right)\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left({B}^{2} \cdot \left(\frac{1}{2} \cdot \frac{{B}^{2} \cdot F}{A} + 2 \cdot \left(A \cdot F\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left({B}^{2}\right), \left(\frac{1}{2} \cdot \frac{{B}^{2} \cdot F}{A} + 2 \cdot \left(A \cdot F\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(B \cdot B\right), \left(\frac{1}{2} \cdot \frac{{B}^{2} \cdot F}{A} + 2 \cdot \left(A \cdot F\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \left(\frac{1}{2} \cdot \frac{{B}^{2} \cdot F}{A} + 2 \cdot \left(A \cdot F\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \frac{{B}^{2} \cdot F}{A}\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{{B}^{2} \cdot F}{A}\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left({B}^{2} \cdot F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({B}^{2}\right), F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(B \cdot B\right), F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), A\right)\right), \mathsf{*.f64}\left(2, \left(A \cdot F\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      12. *-lowering-*.f64 9.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), A\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(A, F\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   10. Simplified 9.4%

       \[\leadsto \frac{\sqrt{\color{blue}{2 \cdot \left(\left(B \cdot B\right) \cdot \left(0.5 \cdot \frac{\left(B \cdot B\right) \cdot F}{A} + 2 \cdot \left(A \cdot F\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   11. Taylor expanded in B around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(4 \cdot \left(A \cdot \left({B}^{2} \cdot F\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(4, \left(A \cdot \left({B}^{2} \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, \left({B}^{2} \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\left({B}^{2}\right), F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\left(B \cdot B\right), F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       5. *-lowering-*.f64 9.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   13. Simplified 9.8%

       \[\leadsto \frac{\sqrt{\color{blue}{4 \cdot \left(A \cdot \left(\left(B \cdot B\right) \cdot F\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
3. Recombined 2 regimes into one program.

4. Final simplification 6.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq 2.65 \cdot 10^{-48}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot \left(B \cdot B\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{4 \cdot \left(A \cdot \left(F \cdot \left(B \cdot B\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 8.8% accurate, 5.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \frac{\sqrt{2 \cdot \left(F \cdot \left(B\_m \cdot \left(B\_m \cdot B\_m\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (/ (sqrt (* 2.0 (* F (* B_m (* B_m B_m))))) (- (* (* 4.0 A) C) (* B_m B_m))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	return sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / (((4.0 * A) * C) - (B_m * B_m));
}

Fortran

B_m = abs(b)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = sqrt((2.0d0 * (f * (b_m * (b_m * b_m))))) / (((4.0d0 * a) * c) - (b_m * b_m))
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	return Math.sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / (((4.0 * A) * C) - (B_m * B_m));
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	return math.sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / (((4.0 * A) * C) - (B_m * B_m))

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	return Float64(sqrt(Float64(2.0 * Float64(F * Float64(B_m * Float64(B_m * B_m))))) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m)))
end

MATLAB

B_m = abs(B);
function tmp = code(A, B_m, C, F)
	tmp = sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / (((4.0 * A) * C) - (B_m * B_m));
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := N[(N[Sqrt[N[(2.0 * N[(F * N[(B$95$m * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.3%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. 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

   \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left({B}^{3} \cdot F\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
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 3.7%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

7. Simplified 3.7%

   \[\leadsto \frac{\sqrt{\color{blue}{2 \cdot \left(\left(B \cdot \left(B \cdot B\right)\right) \cdot F\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

8. Final simplification 3.7%

   \[\leadsto \frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot \left(B \cdot B\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
9. Add Preprocessing

Alternative 25: 3.3% accurate, 5.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ -0.25 \cdot \left(\frac{1}{C} \cdot {\left(\left(B\_m \cdot F\right) \cdot \left(B\_m \cdot F\right)\right)}^{0.25}\right) \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (* -0.25 (* (/ 1.0 C) (pow (* (* B_m F) (* B_m F)) 0.25))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	return -0.25 * ((1.0 / C) * pow(((B_m * F) * (B_m * F)), 0.25));
}

Fortran

B_m = abs(b)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = (-0.25d0) * ((1.0d0 / c) * (((b_m * f) * (b_m * f)) ** 0.25d0))
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	return -0.25 * ((1.0 / C) * Math.pow(((B_m * F) * (B_m * F)), 0.25));
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	return -0.25 * ((1.0 / C) * math.pow(((B_m * F) * (B_m * F)), 0.25))

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	return Float64(-0.25 * Float64(Float64(1.0 / C) * (Float64(Float64(B_m * F) * Float64(B_m * F)) ^ 0.25)))
end

MATLAB

B_m = abs(B);
function tmp = code(A, B_m, C, F)
	tmp = -0.25 * ((1.0 / C) * (((B_m * F) * (B_m * F)) ^ 0.25));
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := N[(-0.25 * N[(N[(1.0 / C), $MachinePrecision] * N[Power[N[(N[(B$95$m * F), $MachinePrecision] * N[(B$95$m * F), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.3%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. 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 2.5%

   \[\leadsto \frac{\sqrt{\color{blue}{\left(B \cdot \left(B \cdot B\right)\right) \cdot \left(2 \cdot \left(F + F \cdot \frac{C + A}{B}\right) + \frac{2 \cdot F}{B} \cdot \frac{-4 \cdot \left(A \cdot C\right) + 0.5 \cdot \left(\left(A - C\right) \cdot \left(A - C\right)\right)}{B}\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

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 1.7%

      \[\leadsto \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(B, F\right)\right), \mathsf{/.f64}\left(1, \color{blue}{C}\right)\right)\right) \]

10. Simplified 1.7%

    \[\leadsto \color{blue}{-0.25 \cdot \left(\sqrt{B \cdot F} \cdot \frac{1}{C}\right)} \]
11. Step-by-step derivation

    1. pow1/2 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(\left({\left(B \cdot F\right)}^{\frac{1}{2}}\right), \mathsf{/.f64}\left(\color{blue}{1}, C\right)\right)\right) \]

    2. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(\left({\left(B \cdot F\right)}^{\left(\frac{1}{4} + \frac{1}{4}\right)}\right), \mathsf{/.f64}\left(1, C\right)\right)\right) \]

    3. pow-prod-up N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(\left({\left(B \cdot F\right)}^{\frac{1}{4}} \cdot {\left(B \cdot F\right)}^{\frac{1}{4}}\right), \mathsf{/.f64}\left(\color{blue}{1}, C\right)\right)\right) \]

    4. pow-prod-down N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(\left({\left(\left(B \cdot F\right) \cdot \left(B \cdot F\right)\right)}^{\frac{1}{4}}\right), \mathsf{/.f64}\left(\color{blue}{1}, C\right)\right)\right) \]

    5. pow-lowering-pow.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\left(B \cdot F\right) \cdot \left(B \cdot F\right)\right), \frac{1}{4}\right), \mathsf{/.f64}\left(\color{blue}{1}, C\right)\right)\right) \]

    6. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(B \cdot F\right), \left(B \cdot F\right)\right), \frac{1}{4}\right), \mathsf{/.f64}\left(1, C\right)\right)\right) \]

    7. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, F\right), \left(B \cdot F\right)\right), \frac{1}{4}\right), \mathsf{/.f64}\left(1, C\right)\right)\right) \]

    8. *-lowering-*.f64 3.3%

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, F\right), \mathsf{*.f64}\left(B, F\right)\right), \frac{1}{4}\right), \mathsf{/.f64}\left(1, C\right)\right)\right) \]

12. Applied egg-rr 3.3%

    \[\leadsto -0.25 \cdot \left(\color{blue}{{\left(\left(B \cdot F\right) \cdot \left(B \cdot F\right)\right)}^{0.25}} \cdot \frac{1}{C}\right) \]

13. Final simplification 3.3%

    \[\leadsto -0.25 \cdot \left(\frac{1}{C} \cdot {\left(\left(B \cdot F\right) \cdot \left(B \cdot F\right)\right)}^{0.25}\right) \]
14. Add Preprocessing

Alternative 26: 3.3% accurate, 5.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \frac{-0.25 \cdot {\left(B\_m \cdot F\right)}^{0.5}}{C} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F) :precision binary64 (/ (* -0.25 (pow (* B_m F) 0.5)) C))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	return (-0.25 * pow((B_m * F), 0.5)) / C;
}

Fortran

B_m = abs(b)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = ((-0.25d0) * ((b_m * f) ** 0.5d0)) / c
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	return (-0.25 * Math.pow((B_m * F), 0.5)) / C;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	return (-0.25 * math.pow((B_m * F), 0.5)) / C

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	return Float64(Float64(-0.25 * (Float64(B_m * F) ^ 0.5)) / C)
end

MATLAB

B_m = abs(B);
function tmp = code(A, B_m, C, F)
	tmp = (-0.25 * ((B_m * F) ^ 0.5)) / C;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := N[(N[(-0.25 * N[Power[N[(B$95$m * F), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]

Derivation

1. Initial program 17.3%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. 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 2.5%

   \[\leadsto \frac{\sqrt{\color{blue}{\left(B \cdot \left(B \cdot B\right)\right) \cdot \left(2 \cdot \left(F + F \cdot \frac{C + A}{B}\right) + \frac{2 \cdot F}{B} \cdot \frac{-4 \cdot \left(A \cdot C\right) + 0.5 \cdot \left(\left(A - C\right) \cdot \left(A - C\right)\right)}{B}\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

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 1.7%

      \[\leadsto \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(B, F\right)\right), \mathsf{/.f64}\left(1, \color{blue}{C}\right)\right)\right) \]

10. Simplified 1.7%

    \[\leadsto \color{blue}{-0.25 \cdot \left(\sqrt{B \cdot F} \cdot \frac{1}{C}\right)} \]
11. Step-by-step derivation

    1. associate-*r* N/A

       \[\leadsto \left(\frac{-1}{4} \cdot \sqrt{B \cdot F}\right) \cdot \color{blue}{\frac{1}{C}} \]

    2. un-div-inv N/A

       \[\leadsto \frac{\frac{-1}{4} \cdot \sqrt{B \cdot F}}{\color{blue}{C}} \]

    3. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{4} \cdot \sqrt{B \cdot F}\right), \color{blue}{C}\right) \]

    4. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \left(\sqrt{B \cdot F}\right)\right), C\right) \]

    5. pow1/2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \left({\left(B \cdot F\right)}^{\frac{1}{2}}\right)\right), C\right) \]

    6. pow-lowering-pow.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{pow.f64}\left(\left(B \cdot F\right), \frac{1}{2}\right)\right), C\right) \]

    7. *-lowering-*.f64 1.9%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(B, F\right), \frac{1}{2}\right)\right), C\right) \]

12. Applied egg-rr 1.9%

    \[\leadsto \color{blue}{\frac{-0.25 \cdot {\left(B \cdot F\right)}^{0.5}}{C}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024155 
(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))))