ABCF->ab-angle a

Percentage Accurate: 19.2% → 54.4%

Time: 24.5s

Alternatives: 12

Speedup: 6.0×

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: 54.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 4 \cdot 10^{-155}:\\ \;\;\;\;\frac{\sqrt{\left(t\_0 \cdot F\right) \cdot \left(C \cdot 4\right)}}{-t\_0}\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{-49}:\\ \;\;\;\;-\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+302}:\\ \;\;\;\;\frac{\left(B\_m \cdot \sqrt{2}\right) \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\_m\right)} \cdot \sqrt{F}\right)}{C \cdot \left(A \cdot 4\right) - {B\_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= (pow B_m 2.0) 4e-155)
     (/ (sqrt (* (* t_0 F) (* C 4.0))) (- t_0))
     (if (<= (pow B_m 2.0) 1e-49)
       (- (sqrt (/ (- F) A)))
       (if (<= (pow B_m 2.0) 2e+302)
         (/
          (* (* B_m (sqrt 2.0)) (* (sqrt (+ C (hypot C B_m))) (sqrt F)))
          (- (* C (* A 4.0)) (pow B_m 2.0)))
         (/ (sqrt (* 2.0 F)) (- (sqrt B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (pow(B_m, 2.0) <= 4e-155) {
		tmp = sqrt(((t_0 * F) * (C * 4.0))) / -t_0;
	} else if (pow(B_m, 2.0) <= 1e-49) {
		tmp = -sqrt((-F / A));
	} else if (pow(B_m, 2.0) <= 2e+302) {
		tmp = ((B_m * sqrt(2.0)) * (sqrt((C + hypot(C, B_m))) * sqrt(F))) / ((C * (A * 4.0)) - pow(B_m, 2.0));
	} else {
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 4e-155)
		tmp = Float64(sqrt(Float64(Float64(t_0 * F) * Float64(C * 4.0))) / Float64(-t_0));
	elseif ((B_m ^ 2.0) <= 1e-49)
		tmp = Float64(-sqrt(Float64(Float64(-F) / A)));
	elseif ((B_m ^ 2.0) <= 2e+302)
		tmp = Float64(Float64(Float64(B_m * sqrt(2.0)) * Float64(sqrt(Float64(C + hypot(C, B_m))) * sqrt(F))) / Float64(Float64(C * Float64(A * 4.0)) - (B_m ^ 2.0)));
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) / Float64(-sqrt(B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e-155], N[(N[Sqrt[N[(N[(t$95$0 * F), $MachinePrecision] * N[(C * 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-49], (-N[Sqrt[N[((-F) / A), $MachinePrecision]], $MachinePrecision]), If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+302], N[(N[(N[(B$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision] - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 4.00000000000000006e-155

   1. Initial program 17.9%

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

   3. Simplified 31.0%

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

   5. Taylor expanded in A around -inf 24.6%

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

   if 4.00000000000000006e-155 < (pow.f64 B #s(literal 2 binary64)) < 9.99999999999999936e-50

   1. Initial program 16.1%

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

   3. Simplified 27.0%

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

   5. Taylor expanded in A around -inf 25.6%

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

   6. Taylor expanded in F around 0 25.3%

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

   7. Taylor expanded in B around 0 29.8%

      \[\leadsto -1 \cdot \sqrt{\color{blue}{-1 \cdot \frac{F}{A}}} \]
   8. Step-by-step derivation

      1. associate-*r/ 29.8%

         \[\leadsto -1 \cdot \sqrt{\color{blue}{\frac{-1 \cdot F}{A}}} \]

      2. neg-mul-1 29.8%

         \[\leadsto -1 \cdot \sqrt{\frac{\color{blue}{-F}}{A}} \]

   9. Simplified 29.8%

      \[\leadsto -1 \cdot \sqrt{\color{blue}{\frac{-F}{A}}} \]

   if 9.99999999999999936e-50 < (pow.f64 B #s(literal 2 binary64)) < 2.0000000000000002e302

   1. Initial program 39.5%

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

   3. Taylor expanded in A around 0 21.2%

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

      1. mul-1-neg 21.2%

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

   5. Simplified 21.2%

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

      1. pow1/2 21.3%

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

      2. *-commutative 21.3%

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

      3. unpow-prod-down 23.7%

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

      4. pow1/2 23.7%

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

      5. +-commutative 23.7%

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

      6. unpow2 23.7%

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

      7. unpow2 23.7%

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

      8. hypot-define 28.1%

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

      9. pow1/2 28.1%

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

   7. Applied egg-rr 28.1%

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

   if 2.0000000000000002e302 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 0.1%

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

   3. Taylor expanded in B around inf 20.8%

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

      1. mul-1-neg 20.8%

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

   5. Simplified 20.8%

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

      1. sqrt-div 41.5%

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

   7. Applied egg-rr 41.5%

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

      1. associate-*l/ 41.5%

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

      2. pow1/2 41.5%

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

      3. pow1/2 41.5%

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

      4. pow-prod-down 41.6%

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

   9. Applied egg-rr 41.6%

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

       1. unpow1/2 41.6%

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

   11. Simplified 41.6%

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

4. Final simplification 30.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 4 \cdot 10^{-155}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(C \cdot 4\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 10^{-49}:\\ \;\;\;\;-\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+302}:\\ \;\;\;\;\frac{\left(B \cdot \sqrt{2}\right) \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right)}{C \cdot \left(A \cdot 4\right) - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 54.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 4 \cdot 10^{-155}:\\ \;\;\;\;\frac{\sqrt{\left(t\_0 \cdot F\right) \cdot \left(C \cdot 4\right)}}{-t\_0}\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{-49}:\\ \;\;\;\;-\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+222}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{F \cdot \frac{A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)}{{B\_m}^{2} + -4 \cdot \left(A \cdot C\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= (pow B_m 2.0) 4e-155)
     (/ (sqrt (* (* t_0 F) (* C 4.0))) (- t_0))
     (if (<= (pow B_m 2.0) 1e-49)
       (- (sqrt (/ (- F) A)))
       (if (<= (pow B_m 2.0) 5e+222)
         (*
          (sqrt 2.0)
          (-
           (sqrt
            (*
             F
             (/
              (+ A (+ C (hypot B_m (- A C))))
              (+ (pow B_m 2.0) (* -4.0 (* A C))))))))
         (/ (sqrt (* 2.0 F)) (- (sqrt B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (pow(B_m, 2.0) <= 4e-155) {
		tmp = sqrt(((t_0 * F) * (C * 4.0))) / -t_0;
	} else if (pow(B_m, 2.0) <= 1e-49) {
		tmp = -sqrt((-F / A));
	} else if (pow(B_m, 2.0) <= 5e+222) {
		tmp = sqrt(2.0) * -sqrt((F * ((A + (C + hypot(B_m, (A - C)))) / (pow(B_m, 2.0) + (-4.0 * (A * C))))));
	} else {
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 4e-155)
		tmp = Float64(sqrt(Float64(Float64(t_0 * F) * Float64(C * 4.0))) / Float64(-t_0));
	elseif ((B_m ^ 2.0) <= 1e-49)
		tmp = Float64(-sqrt(Float64(Float64(-F) / A)));
	elseif ((B_m ^ 2.0) <= 5e+222)
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F * Float64(Float64(A + Float64(C + hypot(B_m, Float64(A - C)))) / Float64((B_m ^ 2.0) + Float64(-4.0 * Float64(A * C))))))));
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) / Float64(-sqrt(B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e-155], N[(N[Sqrt[N[(N[(t$95$0 * F), $MachinePrecision] * N[(C * 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-49], (-N[Sqrt[N[((-F) / A), $MachinePrecision]], $MachinePrecision]), If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+222], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F * N[(N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 4.00000000000000006e-155

   1. Initial program 17.9%

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

   3. Simplified 31.0%

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

   5. Taylor expanded in A around -inf 24.6%

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

   if 4.00000000000000006e-155 < (pow.f64 B #s(literal 2 binary64)) < 9.99999999999999936e-50

   1. Initial program 16.1%

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

   3. Simplified 27.0%

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

   5. Taylor expanded in A around -inf 25.6%

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

   6. Taylor expanded in F around 0 25.3%

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

   7. Taylor expanded in B around 0 29.8%

      \[\leadsto -1 \cdot \sqrt{\color{blue}{-1 \cdot \frac{F}{A}}} \]
   8. Step-by-step derivation

      1. associate-*r/ 29.8%

         \[\leadsto -1 \cdot \sqrt{\color{blue}{\frac{-1 \cdot F}{A}}} \]

      2. neg-mul-1 29.8%

         \[\leadsto -1 \cdot \sqrt{\frac{\color{blue}{-F}}{A}} \]

   9. Simplified 29.8%

      \[\leadsto -1 \cdot \sqrt{\color{blue}{\frac{-F}{A}}} \]

   if 9.99999999999999936e-50 < (pow.f64 B #s(literal 2 binary64)) < 5.00000000000000023e222

   1. Initial program 37.1%

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

   3. Taylor expanded in F around 0 40.3%

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

      1. mul-1-neg 40.3%

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

   5. Simplified 62.4%

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

   if 5.00000000000000023e222 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 9.9%

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

   3. Taylor expanded in B around inf 19.2%

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

      1. mul-1-neg 19.2%

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

   5. Simplified 19.2%

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

      1. sqrt-div 36.5%

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

   7. Applied egg-rr 36.5%

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

      1. associate-*l/ 36.4%

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

      2. pow1/2 36.4%

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

      3. pow1/2 36.4%

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

      4. pow-prod-down 36.5%

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

   9. Applied egg-rr 36.5%

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

       1. unpow1/2 36.5%

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

   11. Simplified 36.5%

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

4. Final simplification 37.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 4 \cdot 10^{-155}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(C \cdot 4\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 10^{-49}:\\ \;\;\;\;-\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+222}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{F \cdot \frac{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}{{B}^{2} + -4 \cdot \left(A \cdot C\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 52.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 4 \cdot 10^{-155}:\\ \;\;\;\;\frac{\sqrt{\left(t\_0 \cdot F\right) \cdot \left(C \cdot 4\right)}}{-t\_0}\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{-49}:\\ \;\;\;\;-\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+302}:\\ \;\;\;\;\frac{-1}{\frac{{B\_m}^{2} - C \cdot \left(A \cdot 4\right)}{B\_m \cdot \sqrt{2 \cdot \left(F \cdot \left(C + \mathsf{hypot}\left(C, B\_m\right)\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= (pow B_m 2.0) 4e-155)
     (/ (sqrt (* (* t_0 F) (* C 4.0))) (- t_0))
     (if (<= (pow B_m 2.0) 1e-49)
       (- (sqrt (/ (- F) A)))
       (if (<= (pow B_m 2.0) 2e+302)
         (/
          -1.0
          (/
           (- (pow B_m 2.0) (* C (* A 4.0)))
           (* B_m (sqrt (* 2.0 (* F (+ C (hypot C B_m))))))))
         (/ (sqrt (* 2.0 F)) (- (sqrt B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (pow(B_m, 2.0) <= 4e-155) {
		tmp = sqrt(((t_0 * F) * (C * 4.0))) / -t_0;
	} else if (pow(B_m, 2.0) <= 1e-49) {
		tmp = -sqrt((-F / A));
	} else if (pow(B_m, 2.0) <= 2e+302) {
		tmp = -1.0 / ((pow(B_m, 2.0) - (C * (A * 4.0))) / (B_m * sqrt((2.0 * (F * (C + hypot(C, B_m)))))));
	} else {
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 4e-155)
		tmp = Float64(sqrt(Float64(Float64(t_0 * F) * Float64(C * 4.0))) / Float64(-t_0));
	elseif ((B_m ^ 2.0) <= 1e-49)
		tmp = Float64(-sqrt(Float64(Float64(-F) / A)));
	elseif ((B_m ^ 2.0) <= 2e+302)
		tmp = Float64(-1.0 / Float64(Float64((B_m ^ 2.0) - Float64(C * Float64(A * 4.0))) / Float64(B_m * sqrt(Float64(2.0 * Float64(F * Float64(C + hypot(C, B_m))))))));
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) / Float64(-sqrt(B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e-155], N[(N[Sqrt[N[(N[(t$95$0 * F), $MachinePrecision] * N[(C * 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-49], (-N[Sqrt[N[((-F) / A), $MachinePrecision]], $MachinePrecision]), If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+302], N[(-1.0 / N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(B$95$m * N[Sqrt[N[(2.0 * N[(F * N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 4.00000000000000006e-155

   1. Initial program 17.9%

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

   3. Simplified 31.0%

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

   5. Taylor expanded in A around -inf 24.6%

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

   if 4.00000000000000006e-155 < (pow.f64 B #s(literal 2 binary64)) < 9.99999999999999936e-50

   1. Initial program 16.1%

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

   3. Simplified 27.0%

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

   5. Taylor expanded in A around -inf 25.6%

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

   6. Taylor expanded in F around 0 25.3%

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

   7. Taylor expanded in B around 0 29.8%

      \[\leadsto -1 \cdot \sqrt{\color{blue}{-1 \cdot \frac{F}{A}}} \]
   8. Step-by-step derivation

      1. associate-*r/ 29.8%

         \[\leadsto -1 \cdot \sqrt{\color{blue}{\frac{-1 \cdot F}{A}}} \]

      2. neg-mul-1 29.8%

         \[\leadsto -1 \cdot \sqrt{\frac{\color{blue}{-F}}{A}} \]

   9. Simplified 29.8%

      \[\leadsto -1 \cdot \sqrt{\color{blue}{\frac{-F}{A}}} \]

   if 9.99999999999999936e-50 < (pow.f64 B #s(literal 2 binary64)) < 2.0000000000000002e302

   1. Initial program 39.5%

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

   3. Taylor expanded in A around 0 21.2%

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

      1. mul-1-neg 21.2%

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

   5. Simplified 21.2%

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

      1. clear-num 21.1%

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

      2. inv-pow 21.1%

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

   7. Applied egg-rr 24.4%

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

      1. unpow-1 24.4%

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

      2. distribute-rgt-neg-in 24.4%

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

      3. unpow1/2 24.4%

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

   9. Simplified 24.4%

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

   if 2.0000000000000002e302 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 0.1%

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

   3. Taylor expanded in B around inf 20.8%

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

      1. mul-1-neg 20.8%

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

   5. Simplified 20.8%

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

      1. sqrt-div 41.5%

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

   7. Applied egg-rr 41.5%

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

      1. associate-*l/ 41.5%

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

      2. pow1/2 41.5%

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

      3. pow1/2 41.5%

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

      4. pow-prod-down 41.6%

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

   9. Applied egg-rr 41.6%

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

       1. unpow1/2 41.6%

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

   11. Simplified 41.6%

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

4. Final simplification 29.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 4 \cdot 10^{-155}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(C \cdot 4\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 10^{-49}:\\ \;\;\;\;-\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+302}:\\ \;\;\;\;\frac{-1}{\frac{{B}^{2} - C \cdot \left(A \cdot 4\right)}{B \cdot \sqrt{2 \cdot \left(F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 49.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 10^{-49}:\\ \;\;\;\;-\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+302}:\\ \;\;\;\;\frac{-1}{\frac{{B\_m}^{2} - C \cdot \left(A \cdot 4\right)}{B\_m \cdot \sqrt{2 \cdot \left(F \cdot \left(C + \mathsf{hypot}\left(C, B\_m\right)\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 1e-49)
   (- (sqrt (/ (- F) A)))
   (if (<= (pow B_m 2.0) 2e+302)
     (/
      -1.0
      (/
       (- (pow B_m 2.0) (* C (* A 4.0)))
       (* B_m (sqrt (* 2.0 (* F (+ C (hypot C B_m))))))))
     (/ (sqrt (* 2.0 F)) (- (sqrt B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 1e-49) {
		tmp = -sqrt((-F / A));
	} else if (pow(B_m, 2.0) <= 2e+302) {
		tmp = -1.0 / ((pow(B_m, 2.0) - (C * (A * 4.0))) / (B_m * sqrt((2.0 * (F * (C + hypot(C, B_m)))))));
	} else {
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (Math.pow(B_m, 2.0) <= 1e-49) {
		tmp = -Math.sqrt((-F / A));
	} else if (Math.pow(B_m, 2.0) <= 2e+302) {
		tmp = -1.0 / ((Math.pow(B_m, 2.0) - (C * (A * 4.0))) / (B_m * Math.sqrt((2.0 * (F * (C + Math.hypot(C, B_m)))))));
	} else {
		tmp = Math.sqrt((2.0 * F)) / -Math.sqrt(B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if math.pow(B_m, 2.0) <= 1e-49:
		tmp = -math.sqrt((-F / A))
	elif math.pow(B_m, 2.0) <= 2e+302:
		tmp = -1.0 / ((math.pow(B_m, 2.0) - (C * (A * 4.0))) / (B_m * math.sqrt((2.0 * (F * (C + math.hypot(C, B_m)))))))
	else:
		tmp = math.sqrt((2.0 * F)) / -math.sqrt(B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-49)
		tmp = Float64(-sqrt(Float64(Float64(-F) / A)));
	elseif ((B_m ^ 2.0) <= 2e+302)
		tmp = Float64(-1.0 / Float64(Float64((B_m ^ 2.0) - Float64(C * Float64(A * 4.0))) / Float64(B_m * sqrt(Float64(2.0 * Float64(F * Float64(C + hypot(C, B_m))))))));
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) / Float64(-sqrt(B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 1e-49)
		tmp = -sqrt((-F / A));
	elseif ((B_m ^ 2.0) <= 2e+302)
		tmp = -1.0 / (((B_m ^ 2.0) - (C * (A * 4.0))) / (B_m * sqrt((2.0 * (F * (C + hypot(C, B_m)))))));
	else
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-49], (-N[Sqrt[N[((-F) / A), $MachinePrecision]], $MachinePrecision]), If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+302], N[(-1.0 / N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(B$95$m * N[Sqrt[N[(2.0 * N[(F * N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.99999999999999936e-50

   1. Initial program 17.6%

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

   3. Simplified 30.3%

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

   5. Taylor expanded in A around -inf 24.6%

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

   6. Taylor expanded in F around 0 15.1%

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

   7. Taylor expanded in B around 0 16.8%

      \[\leadsto -1 \cdot \sqrt{\color{blue}{-1 \cdot \frac{F}{A}}} \]
   8. Step-by-step derivation

      1. associate-*r/ 16.8%

         \[\leadsto -1 \cdot \sqrt{\color{blue}{\frac{-1 \cdot F}{A}}} \]

      2. neg-mul-1 16.8%

         \[\leadsto -1 \cdot \sqrt{\frac{\color{blue}{-F}}{A}} \]

   9. Simplified 16.8%

      \[\leadsto -1 \cdot \sqrt{\color{blue}{\frac{-F}{A}}} \]

   if 9.99999999999999936e-50 < (pow.f64 B #s(literal 2 binary64)) < 2.0000000000000002e302

   1. Initial program 39.5%

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

   3. Taylor expanded in A around 0 21.2%

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

      1. mul-1-neg 21.2%

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

   5. Simplified 21.2%

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

      1. clear-num 21.1%

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

      2. inv-pow 21.1%

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

   7. Applied egg-rr 24.4%

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

      1. unpow-1 24.4%

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

      2. distribute-rgt-neg-in 24.4%

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

      3. unpow1/2 24.4%

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

   9. Simplified 24.4%

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

   if 2.0000000000000002e302 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 0.1%

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

   3. Taylor expanded in B around inf 20.8%

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

      1. mul-1-neg 20.8%

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

   5. Simplified 20.8%

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

      1. sqrt-div 41.5%

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

   7. Applied egg-rr 41.5%

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

      1. associate-*l/ 41.5%

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

      2. pow1/2 41.5%

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

      3. pow1/2 41.5%

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

      4. pow-prod-down 41.6%

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

   9. Applied egg-rr 41.6%

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

       1. unpow1/2 41.6%

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

   11. Simplified 41.6%

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

4. Final simplification 25.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-49}:\\ \;\;\;\;-\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+302}:\\ \;\;\;\;\frac{-1}{\frac{{B}^{2} - C \cdot \left(A \cdot 4\right)}{B \cdot \sqrt{2 \cdot \left(F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 49.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 10^{-49}:\\ \;\;\;\;-\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+302}:\\ \;\;\;\;\frac{B\_m \cdot \sqrt{2 \cdot \left(F \cdot \left(C + \mathsf{hypot}\left(C, B\_m\right)\right)\right)}}{C \cdot \left(A \cdot 4\right) - {B\_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 1e-49)
   (- (sqrt (/ (- F) A)))
   (if (<= (pow B_m 2.0) 2e+302)
     (/
      (* B_m (sqrt (* 2.0 (* F (+ C (hypot C B_m))))))
      (- (* C (* A 4.0)) (pow B_m 2.0)))
     (/ (sqrt (* 2.0 F)) (- (sqrt B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 1e-49) {
		tmp = -sqrt((-F / A));
	} else if (pow(B_m, 2.0) <= 2e+302) {
		tmp = (B_m * sqrt((2.0 * (F * (C + hypot(C, B_m)))))) / ((C * (A * 4.0)) - pow(B_m, 2.0));
	} else {
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (Math.pow(B_m, 2.0) <= 1e-49) {
		tmp = -Math.sqrt((-F / A));
	} else if (Math.pow(B_m, 2.0) <= 2e+302) {
		tmp = (B_m * Math.sqrt((2.0 * (F * (C + Math.hypot(C, B_m)))))) / ((C * (A * 4.0)) - Math.pow(B_m, 2.0));
	} else {
		tmp = Math.sqrt((2.0 * F)) / -Math.sqrt(B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if math.pow(B_m, 2.0) <= 1e-49:
		tmp = -math.sqrt((-F / A))
	elif math.pow(B_m, 2.0) <= 2e+302:
		tmp = (B_m * math.sqrt((2.0 * (F * (C + math.hypot(C, B_m)))))) / ((C * (A * 4.0)) - math.pow(B_m, 2.0))
	else:
		tmp = math.sqrt((2.0 * F)) / -math.sqrt(B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-49)
		tmp = Float64(-sqrt(Float64(Float64(-F) / A)));
	elseif ((B_m ^ 2.0) <= 2e+302)
		tmp = Float64(Float64(B_m * sqrt(Float64(2.0 * Float64(F * Float64(C + hypot(C, B_m)))))) / Float64(Float64(C * Float64(A * 4.0)) - (B_m ^ 2.0)));
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) / Float64(-sqrt(B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 1e-49)
		tmp = -sqrt((-F / A));
	elseif ((B_m ^ 2.0) <= 2e+302)
		tmp = (B_m * sqrt((2.0 * (F * (C + hypot(C, B_m)))))) / ((C * (A * 4.0)) - (B_m ^ 2.0));
	else
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-49], (-N[Sqrt[N[((-F) / A), $MachinePrecision]], $MachinePrecision]), If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+302], N[(N[(B$95$m * N[Sqrt[N[(2.0 * N[(F * N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision] - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.99999999999999936e-50

   1. Initial program 17.6%

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

   3. Simplified 30.3%

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

   5. Taylor expanded in A around -inf 24.6%

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

   6. Taylor expanded in F around 0 15.1%

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

   7. Taylor expanded in B around 0 16.8%

      \[\leadsto -1 \cdot \sqrt{\color{blue}{-1 \cdot \frac{F}{A}}} \]
   8. Step-by-step derivation

      1. associate-*r/ 16.8%

         \[\leadsto -1 \cdot \sqrt{\color{blue}{\frac{-1 \cdot F}{A}}} \]

      2. neg-mul-1 16.8%

         \[\leadsto -1 \cdot \sqrt{\frac{\color{blue}{-F}}{A}} \]

   9. Simplified 16.8%

      \[\leadsto -1 \cdot \sqrt{\color{blue}{\frac{-F}{A}}} \]

   if 9.99999999999999936e-50 < (pow.f64 B #s(literal 2 binary64)) < 2.0000000000000002e302

   1. Initial program 39.5%

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

   3. Taylor expanded in A around 0 21.2%

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

      1. mul-1-neg 21.2%

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

   5. Simplified 21.2%

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

      1. *-un-lft-identity 21.2%

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

   7. Applied egg-rr 24.4%

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

      1. *-lft-identity 24.4%

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

      2. distribute-frac-neg 24.4%

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

      3. distribute-neg-frac2 24.4%

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

      4. unpow1/2 24.3%

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

   9. Simplified 24.3%

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

   if 2.0000000000000002e302 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 0.1%

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

   3. Taylor expanded in B around inf 20.8%

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

      1. mul-1-neg 20.8%

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

   5. Simplified 20.8%

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

      1. sqrt-div 41.5%

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

   7. Applied egg-rr 41.5%

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

      1. associate-*l/ 41.5%

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

      2. pow1/2 41.5%

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

      3. pow1/2 41.5%

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

      4. pow-prod-down 41.6%

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

   9. Applied egg-rr 41.6%

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

       1. unpow1/2 41.6%

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

   11. Simplified 41.6%

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

4. Final simplification 25.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-49}:\\ \;\;\;\;-\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+302}:\\ \;\;\;\;\frac{B \cdot \sqrt{2 \cdot \left(F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 49.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 10^{-49}:\\ \;\;\;\;-\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+292}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B\_m, C\right)\right)} \cdot \frac{\sqrt{2}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 1e-49)
   (- (sqrt (/ (- F) A)))
   (if (<= (pow B_m 2.0) 5e+292)
     (* (sqrt (* F (+ C (hypot B_m C)))) (/ (sqrt 2.0) (- B_m)))
     (/ (sqrt (* 2.0 F)) (- (sqrt B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 1e-49) {
		tmp = -sqrt((-F / A));
	} else if (pow(B_m, 2.0) <= 5e+292) {
		tmp = sqrt((F * (C + hypot(B_m, C)))) * (sqrt(2.0) / -B_m);
	} else {
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (Math.pow(B_m, 2.0) <= 1e-49) {
		tmp = -Math.sqrt((-F / A));
	} else if (Math.pow(B_m, 2.0) <= 5e+292) {
		tmp = Math.sqrt((F * (C + Math.hypot(B_m, C)))) * (Math.sqrt(2.0) / -B_m);
	} else {
		tmp = Math.sqrt((2.0 * F)) / -Math.sqrt(B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if math.pow(B_m, 2.0) <= 1e-49:
		tmp = -math.sqrt((-F / A))
	elif math.pow(B_m, 2.0) <= 5e+292:
		tmp = math.sqrt((F * (C + math.hypot(B_m, C)))) * (math.sqrt(2.0) / -B_m)
	else:
		tmp = math.sqrt((2.0 * F)) / -math.sqrt(B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-49)
		tmp = Float64(-sqrt(Float64(Float64(-F) / A)));
	elseif ((B_m ^ 2.0) <= 5e+292)
		tmp = Float64(sqrt(Float64(F * Float64(C + hypot(B_m, C)))) * Float64(sqrt(2.0) / Float64(-B_m)));
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) / Float64(-sqrt(B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 1e-49)
		tmp = -sqrt((-F / A));
	elseif ((B_m ^ 2.0) <= 5e+292)
		tmp = sqrt((F * (C + hypot(B_m, C)))) * (sqrt(2.0) / -B_m);
	else
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-49], (-N[Sqrt[N[((-F) / A), $MachinePrecision]], $MachinePrecision]), If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+292], N[(N[Sqrt[N[(F * N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.99999999999999936e-50

   1. Initial program 17.6%

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

   3. Simplified 30.3%

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

   5. Taylor expanded in A around -inf 24.6%

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

   6. Taylor expanded in F around 0 15.1%

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

   7. Taylor expanded in B around 0 16.8%

      \[\leadsto -1 \cdot \sqrt{\color{blue}{-1 \cdot \frac{F}{A}}} \]
   8. Step-by-step derivation

      1. associate-*r/ 16.8%

         \[\leadsto -1 \cdot \sqrt{\color{blue}{\frac{-1 \cdot F}{A}}} \]

      2. neg-mul-1 16.8%

         \[\leadsto -1 \cdot \sqrt{\frac{\color{blue}{-F}}{A}} \]

   9. Simplified 16.8%

      \[\leadsto -1 \cdot \sqrt{\color{blue}{\frac{-F}{A}}} \]

   if 9.99999999999999936e-50 < (pow.f64 B #s(literal 2 binary64)) < 4.9999999999999996e292

   1. Initial program 40.1%

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

   3. Taylor expanded in A around 0 21.5%

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

      1. mul-1-neg 21.5%

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

   5. Simplified 21.5%

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

   6. Taylor expanded in A around 0 21.6%

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

      1. mul-1-neg 21.6%

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

      2. unpow2 21.6%

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

      3. unpow2 21.6%

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

      4. hypot-undefine 24.6%

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

   8. Simplified 24.6%

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

   if 4.9999999999999996e292 < (pow.f64 B #s(literal 2 binary64))

   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 20.5%

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

      1. mul-1-neg 20.5%

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

   5. Simplified 20.5%

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

      1. sqrt-div 40.9%

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

   7. Applied egg-rr 40.9%

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

      1. associate-*l/ 40.9%

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

      2. pow1/2 40.9%

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

      3. pow1/2 40.9%

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

      4. pow-prod-down 41.0%

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

   9. Applied egg-rr 41.0%

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

       1. unpow1/2 41.0%

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

   11. Simplified 41.0%

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

4. Final simplification 25.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-49}:\\ \;\;\;\;-\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+292}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)} \cdot \frac{\sqrt{2}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 47.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 10^{-49}:\\ \;\;\;\;-\sqrt{\frac{-F}{A}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 1e-49)
   (- (sqrt (/ (- F) A)))
   (/ (sqrt (* 2.0 F)) (- (sqrt B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 1e-49) {
		tmp = -sqrt((-F / A));
	} else {
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
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 ** 2.0d0) <= 1d-49) then
        tmp = -sqrt((-f / a))
    else
        tmp = sqrt((2.0d0 * f)) / -sqrt(b_m)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (Math.pow(B_m, 2.0) <= 1e-49) {
		tmp = -Math.sqrt((-F / A));
	} else {
		tmp = Math.sqrt((2.0 * F)) / -Math.sqrt(B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if math.pow(B_m, 2.0) <= 1e-49:
		tmp = -math.sqrt((-F / A))
	else:
		tmp = math.sqrt((2.0 * F)) / -math.sqrt(B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-49)
		tmp = Float64(-sqrt(Float64(Float64(-F) / A)));
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) / Float64(-sqrt(B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 1e-49)
		tmp = -sqrt((-F / A));
	else
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-49], (-N[Sqrt[N[((-F) / A), $MachinePrecision]], $MachinePrecision]), N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.99999999999999936e-50

   1. Initial program 17.6%

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

   3. Simplified 30.3%

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

   5. Taylor expanded in A around -inf 24.6%

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

   6. Taylor expanded in F around 0 15.1%

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

   7. Taylor expanded in B around 0 16.8%

      \[\leadsto -1 \cdot \sqrt{\color{blue}{-1 \cdot \frac{F}{A}}} \]
   8. Step-by-step derivation

      1. associate-*r/ 16.8%

         \[\leadsto -1 \cdot \sqrt{\color{blue}{\frac{-1 \cdot F}{A}}} \]

      2. neg-mul-1 16.8%

         \[\leadsto -1 \cdot \sqrt{\frac{\color{blue}{-F}}{A}} \]

   9. Simplified 16.8%

      \[\leadsto -1 \cdot \sqrt{\color{blue}{\frac{-F}{A}}} \]

   if 9.99999999999999936e-50 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 21.2%

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

   3. Taylor expanded in B around inf 20.3%

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

      1. mul-1-neg 20.3%

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

   5. Simplified 20.3%

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

      1. sqrt-div 31.0%

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

   7. Applied egg-rr 31.0%

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

      1. associate-*l/ 30.9%

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

      2. pow1/2 30.9%

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

      3. pow1/2 30.9%

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

      4. pow-prod-down 31.0%

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

   9. Applied egg-rr 31.0%

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

       1. unpow1/2 31.0%

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

   11. Simplified 31.0%

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

4. Final simplification 24.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-49}:\\ \;\;\;\;-\sqrt{\frac{-F}{A}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 40.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 1.15 \cdot 10^{-24}:\\ \;\;\;\;-\sqrt{\frac{-F}{A}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\left|F \cdot \frac{2}{B\_m}\right|}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 1.15e-24)
   (- (sqrt (/ (- F) A)))
   (- (sqrt (fabs (* F (/ 2.0 B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.15e-24) {
		tmp = -sqrt((-F / A));
	} else {
		tmp = -sqrt(fabs((F * (2.0 / B_m))));
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
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-24) then
        tmp = -sqrt((-f / a))
    else
        tmp = -sqrt(abs((f * (2.0d0 / b_m))))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.15e-24) {
		tmp = -Math.sqrt((-F / A));
	} else {
		tmp = -Math.sqrt(Math.abs((F * (2.0 / B_m))));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 1.15e-24:
		tmp = -math.sqrt((-F / A))
	else:
		tmp = -math.sqrt(math.fabs((F * (2.0 / B_m))))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 1.15e-24)
		tmp = Float64(-sqrt(Float64(Float64(-F) / A)));
	else
		tmp = Float64(-sqrt(abs(Float64(F * Float64(2.0 / B_m)))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 1.15e-24)
		tmp = -sqrt((-F / A));
	else
		tmp = -sqrt(abs((F * (2.0 / B_m))));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 1.15e-24], (-N[Sqrt[N[((-F) / A), $MachinePrecision]], $MachinePrecision]), (-N[Sqrt[N[Abs[N[(F * N[(2.0 / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if B < 1.1500000000000001e-24

   1. Initial program 20.2%

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

   3. Simplified 29.1%

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

   5. Taylor expanded in A around -inf 16.7%

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

   6. Taylor expanded in F around 0 10.9%

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

   7. Taylor expanded in B around 0 13.6%

      \[\leadsto -1 \cdot \sqrt{\color{blue}{-1 \cdot \frac{F}{A}}} \]
   8. Step-by-step derivation

      1. associate-*r/ 13.6%

         \[\leadsto -1 \cdot \sqrt{\color{blue}{\frac{-1 \cdot F}{A}}} \]

      2. neg-mul-1 13.6%

         \[\leadsto -1 \cdot \sqrt{\frac{\color{blue}{-F}}{A}} \]

   9. Simplified 13.6%

      \[\leadsto -1 \cdot \sqrt{\color{blue}{\frac{-F}{A}}} \]

   if 1.1500000000000001e-24 < B

   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. Add Preprocessing

   3. Taylor expanded in B around inf 39.3%

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

      1. mul-1-neg 39.3%

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

   5. Simplified 39.3%

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

      1. neg-sub0 39.3%

         \[\leadsto \color{blue}{0 - \sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

      2. sqrt-unprod 39.5%

         \[\leadsto 0 - \color{blue}{\sqrt{\frac{F}{B} \cdot 2}} \]

   7. Applied egg-rr 39.5%

      \[\leadsto \color{blue}{0 - \sqrt{\frac{F}{B} \cdot 2}} \]
   8. Step-by-step derivation

      1. neg-sub0 39.5%

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

   9. Simplified 39.5%

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

       1. add-sqr-sqrt 39.5%

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

       2. pow1/2 39.5%

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

       3. pow1/2 39.5%

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

       4. pow-prod-down 29.1%

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

       5. pow2 29.1%

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

       6. *-commutative 29.1%

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

   11. Applied egg-rr 29.1%

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

       1. unpow1/2 29.1%

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

       2. unpow2 29.1%

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

       3. rem-sqrt-square 39.5%

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

       4. associate-*r/ 39.5%

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

       5. *-commutative 39.5%

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

       6. associate-/l* 39.5%

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

   13. Simplified 39.5%

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

4. Final simplification 20.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.15 \cdot 10^{-24}:\\ \;\;\;\;-\sqrt{\frac{-F}{A}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\left|F \cdot \frac{2}{B}\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 40.1% accurate, 5.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 6 \cdot 10^{-25}:\\ \;\;\;\;-\sqrt{\frac{-F}{A}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{F}{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 6e-25) (- (sqrt (/ (- F) A))) (- (sqrt (* 2.0 (/ F B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 6e-25) {
		tmp = -sqrt((-F / A));
	} else {
		tmp = -sqrt((2.0 * (F / B_m)));
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
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 <= 6d-25) then
        tmp = -sqrt((-f / a))
    else
        tmp = -sqrt((2.0d0 * (f / b_m)))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 6e-25) {
		tmp = -Math.sqrt((-F / A));
	} else {
		tmp = -Math.sqrt((2.0 * (F / B_m)));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 6e-25:
		tmp = -math.sqrt((-F / A))
	else:
		tmp = -math.sqrt((2.0 * (F / B_m)))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 6e-25)
		tmp = Float64(-sqrt(Float64(Float64(-F) / A)));
	else
		tmp = Float64(-sqrt(Float64(2.0 * Float64(F / B_m))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 6e-25)
		tmp = -sqrt((-F / A));
	else
		tmp = -sqrt((2.0 * (F / B_m)));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 6e-25], (-N[Sqrt[N[((-F) / A), $MachinePrecision]], $MachinePrecision]), (-N[Sqrt[N[(2.0 * N[(F / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if B < 5.9999999999999995e-25

   1. Initial program 20.2%

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

   3. Simplified 29.1%

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

   5. Taylor expanded in A around -inf 16.7%

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

   6. Taylor expanded in F around 0 10.9%

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

   7. Taylor expanded in B around 0 13.6%

      \[\leadsto -1 \cdot \sqrt{\color{blue}{-1 \cdot \frac{F}{A}}} \]
   8. Step-by-step derivation

      1. associate-*r/ 13.6%

         \[\leadsto -1 \cdot \sqrt{\color{blue}{\frac{-1 \cdot F}{A}}} \]

      2. neg-mul-1 13.6%

         \[\leadsto -1 \cdot \sqrt{\frac{\color{blue}{-F}}{A}} \]

   9. Simplified 13.6%

      \[\leadsto -1 \cdot \sqrt{\color{blue}{\frac{-F}{A}}} \]

   if 5.9999999999999995e-25 < B

   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. Add Preprocessing

   3. Taylor expanded in B around inf 39.3%

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

      1. mul-1-neg 39.3%

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

   5. Simplified 39.3%

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

      1. neg-sub0 39.3%

         \[\leadsto \color{blue}{0 - \sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

      2. sqrt-unprod 39.5%

         \[\leadsto 0 - \color{blue}{\sqrt{\frac{F}{B} \cdot 2}} \]

   7. Applied egg-rr 39.5%

      \[\leadsto \color{blue}{0 - \sqrt{\frac{F}{B} \cdot 2}} \]
   8. Step-by-step derivation

      1. neg-sub0 39.5%

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

   9. Simplified 39.5%

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

4. Final simplification 20.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 6 \cdot 10^{-25}:\\ \;\;\;\;-\sqrt{\frac{-F}{A}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{F}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 26.6% accurate, 6.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ -\sqrt{2 \cdot \frac{F}{B\_m}} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (- (sqrt (* 2.0 (/ F B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return -sqrt((2.0 * (F / B_m)));
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = -sqrt((2.0d0 * (f / b_m)))
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	return -Math.sqrt((2.0 * (F / B_m)));
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return -math.sqrt((2.0 * (F / B_m)))

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(-sqrt(Float64(2.0 * Float64(F / B_m))))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = -sqrt((2.0 * (F / B_m)));
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := (-N[Sqrt[N[(2.0 * N[(F / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])

Derivation

1. Initial program 19.6%

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

3. Taylor expanded in B around inf 13.3%

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

   1. mul-1-neg 13.3%

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

5. Simplified 13.3%

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

   1. neg-sub0 13.3%

      \[\leadsto \color{blue}{0 - \sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

   2. sqrt-unprod 13.3%

      \[\leadsto 0 - \color{blue}{\sqrt{\frac{F}{B} \cdot 2}} \]

7. Applied egg-rr 13.3%

   \[\leadsto \color{blue}{0 - \sqrt{\frac{F}{B} \cdot 2}} \]
8. Step-by-step derivation

   1. neg-sub0 13.3%

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

9. Simplified 13.3%

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

10. Final simplification 13.3%

    \[\leadsto -\sqrt{2 \cdot \frac{F}{B}} \]
11. Add Preprocessing

Alternative 11: 26.6% accurate, 6.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ -\sqrt{F \cdot \frac{2}{B\_m}} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (- (sqrt (* F (/ 2.0 B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return -sqrt((F * (2.0 / B_m)));
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
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((f * (2.0d0 / b_m)))
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	return -Math.sqrt((F * (2.0 / B_m)));
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return -math.sqrt((F * (2.0 / B_m)))

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(-sqrt(Float64(F * Float64(2.0 / B_m))))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = -sqrt((F * (2.0 / B_m)));
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := (-N[Sqrt[N[(F * N[(2.0 / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])

Derivation

1. Initial program 19.6%

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

3. Taylor expanded in B around inf 13.3%

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

   1. mul-1-neg 13.3%

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

5. Simplified 13.3%

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

   1. neg-sub0 13.3%

      \[\leadsto \color{blue}{0 - \sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

   2. sqrt-unprod 13.3%

      \[\leadsto 0 - \color{blue}{\sqrt{\frac{F}{B} \cdot 2}} \]

7. Applied egg-rr 13.3%

   \[\leadsto \color{blue}{0 - \sqrt{\frac{F}{B} \cdot 2}} \]
8. Step-by-step derivation

   1. neg-sub0 13.3%

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

9. Simplified 13.3%

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

10. Taylor expanded in F around 0 13.3%

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

    1. associate-*r/ 13.3%

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

    2. *-commutative 13.3%

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

    3. associate-/l* 13.3%

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

12. Simplified 13.3%

    \[\leadsto -\sqrt{\color{blue}{F \cdot \frac{2}{B}}} \]
13. Add Preprocessing

Alternative 12: 2.4% accurate, 6.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \sqrt{F \cdot \frac{2}{B\_m}} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (sqrt (* F (/ 2.0 B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return sqrt((F * (2.0 / B_m)));
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
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((f * (2.0d0 / b_m)))
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	return Math.sqrt((F * (2.0 / B_m)));
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return math.sqrt((F * (2.0 / B_m)))

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return sqrt(Float64(F * Float64(2.0 / B_m)))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = sqrt((F * (2.0 / B_m)));
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := N[Sqrt[N[(F * N[(2.0 / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 19.6%

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

3. Taylor expanded in B around inf 13.3%

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

   1. mul-1-neg 13.3%

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

5. Simplified 13.3%

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

   1. sqrt-div 19.3%

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

7. Applied egg-rr 19.3%

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

   1. associate-*l/ 19.3%

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

   2. pow1/2 19.3%

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

   3. pow1/2 19.3%

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

   4. pow-prod-down 19.3%

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

9. Applied egg-rr 19.3%

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

    1. unpow1/2 19.3%

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

11. Simplified 19.3%

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

    1. sqrt-undiv 13.3%

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

    2. associate-*r/ 13.3%

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

    3. add-sqr-sqrt 0.9%

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

    4. sqrt-unprod 2.1%

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

    5. sqr-neg 2.1%

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

    6. add-sqr-sqrt 2.1%

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

    7. *-un-lft-identity 2.1%

       \[\leadsto \color{blue}{1 \cdot \sqrt{F \cdot \frac{2}{B}}} \]

    8. associate-*r/ 2.1%

       \[\leadsto 1 \cdot \sqrt{\color{blue}{\frac{F \cdot 2}{B}}} \]

13. Applied egg-rr 2.1%

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

    1. *-lft-identity 2.1%

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

    2. associate-/l* 2.1%

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

15. Simplified 2.1%

    \[\leadsto \color{blue}{\sqrt{F \cdot \frac{2}{B}}} \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024136 
(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))))