ABCF->ab-angle b

Percentage Accurate: 19.0% → 46.1%

Time: 30.4s

Alternatives: 18

Speedup: 4.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 19.0% 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: 46.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - \left(A \cdot C\right) \cdot 4\\ t_1 := -\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}\\ \mathbf{if}\;B \leq -1.2 \cdot 10^{+145}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;B \leq -1.52 \cdot 10^{-55}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \left(\sqrt{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \cdot t_1\right)}{t_0}\\ \mathbf{elif}\;B \leq 2.5 \cdot 10^{-96}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(2 \cdot A\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot t_1\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* (* A C) 4.0)))
        (t_1 (- (sqrt (* F (- A (hypot A B)))))))
   (if (<= B -1.2e+145)
     (* (sqrt 2.0) (- (sqrt (/ F B))))
     (if (<= B -1.52e-55)
       (/ (* (sqrt 2.0) (* (sqrt (fma B B (* -4.0 (* A C)))) t_1)) t_0)
       (if (<= B 2.5e-96)
         (/ (- (sqrt (* 2.0 (* (* F t_0) (* 2.0 A))))) t_0)
         (* (/ (sqrt 2.0) B) t_1))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - ((A * C) * 4.0);
	double t_1 = -sqrt((F * (A - hypot(A, B))));
	double tmp;
	if (B <= -1.2e+145) {
		tmp = sqrt(2.0) * -sqrt((F / B));
	} else if (B <= -1.52e-55) {
		tmp = (sqrt(2.0) * (sqrt(fma(B, B, (-4.0 * (A * C)))) * t_1)) / t_0;
	} else if (B <= 2.5e-96) {
		tmp = -sqrt((2.0 * ((F * t_0) * (2.0 * A)))) / t_0;
	} else {
		tmp = (sqrt(2.0) / B) * t_1;
	}
	return tmp;
}

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(Float64(A * C) * 4.0))
	t_1 = Float64(-sqrt(Float64(F * Float64(A - hypot(A, B)))))
	tmp = 0.0
	if (B <= -1.2e+145)
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F / B))));
	elseif (B <= -1.52e-55)
		tmp = Float64(Float64(sqrt(2.0) * Float64(sqrt(fma(B, B, Float64(-4.0 * Float64(A * C)))) * t_1)) / t_0);
	elseif (B <= 2.5e-96)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(F * t_0) * Float64(2.0 * A))))) / t_0);
	else
		tmp = Float64(Float64(sqrt(2.0) / B) * t_1);
	end
	return tmp
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(N[(A * C), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[Sqrt[N[(F * N[(A - N[Sqrt[A ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])}, If[LessEqual[B, -1.2e+145], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[B, -1.52e-55], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[N[(B * B + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[B, 2.5e-96], N[((-N[Sqrt[N[(2.0 * N[(N[(F * t$95$0), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < -1.19999999999999996e145

   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. Step-by-step derivation

   3. Simplified 0.0%

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

   4. Taylor expanded in C around 0 0.0%

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

      1. +-commutative 0.0%

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

      2. unpow2 0.0%

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

      3. unpow2 0.0%

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

      4. hypot-def 0.0%

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

   6. Simplified 0.0%

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

   7. Taylor expanded in B around -inf 0.0%

      \[\leadsto \frac{-\sqrt{2 \cdot \color{blue}{\left(F \cdot {B}^{3}\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   8. Taylor expanded in A around 0 49.5%

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

      1. mul-1-neg 49.5%

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

      2. distribute-rgt-neg-in 49.5%

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

   10. Simplified 49.5%

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

   if -1.19999999999999996e145 < B < -1.5200000000000001e-55

   1. Initial program 36.4%

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

   3. Simplified 36.4%

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

   4. Taylor expanded in C around 0 35.1%

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

      1. +-commutative 35.1%

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

      2. unpow2 35.1%

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

      3. unpow2 35.1%

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

      4. hypot-def 37.0%

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

   6. Simplified 37.0%

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

      1. sqrt-prod 37.0%

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

      2. associate-*l* 37.1%

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

      3. cancel-sign-sub-inv 37.1%

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

      4. metadata-eval 37.1%

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

   8. Applied egg-rr 37.1%

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

      1. sqrt-prod 38.4%

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

      2. fma-def 38.4%

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

   10. Applied egg-rr 38.4%

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

   if -1.5200000000000001e-55 < B < 2.49999999999999997e-96

   1. Initial program 15.1%

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

   3. Simplified 15.1%

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

   4. Taylor expanded in A around -inf 22.3%

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

   if 2.49999999999999997e-96 < B

   1. Initial program 21.2%

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

   3. Simplified 21.2%

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

   4. Taylor expanded in C around 0 22.7%

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

      1. mul-1-neg 22.7%

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

      2. +-commutative 22.7%

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

      3. unpow2 22.7%

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

      4. unpow2 22.7%

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

      5. hypot-def 49.4%

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

   6. Simplified 49.4%

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

4. Final simplification 38.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -1.2 \cdot 10^{+145}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;B \leq -1.52 \cdot 10^{-55}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \left(\sqrt{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}\right)\right)}{B \cdot B - \left(A \cdot C\right) \cdot 4}\\ \mathbf{elif}\;B \leq 2.5 \cdot 10^{-96}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - \left(A \cdot C\right) \cdot 4\right)\right) \cdot \left(2 \cdot A\right)\right)}}{B \cdot B - \left(A \cdot C\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}\right)\\ \end{array} \]

Alternative 2: 45.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)\\ t_1 := B \cdot B - \left(A \cdot C\right) \cdot 4\\ \mathbf{if}\;B \leq -7.6 \cdot 10^{+78}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;B \leq -1.32 \cdot 10^{-55}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 2.5 \cdot 10^{-96}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_1\right) \cdot \left(2 \cdot A\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma B B (* C (* -4.0 A)))) (t_1 (- (* B B) (* (* A C) 4.0))))
   (if (<= B -7.6e+78)
     (* (sqrt 2.0) (- (sqrt (/ F B))))
     (if (<= B -1.32e-55)
       (/ (- (sqrt (* 2.0 (* (* F t_0) (- A (- (hypot B (- A C)) C)))))) t_0)
       (if (<= B 2.5e-96)
         (/ (- (sqrt (* 2.0 (* (* F t_1) (* 2.0 A))))) t_1)
         (* (/ (sqrt 2.0) B) (- (sqrt (* F (- A (hypot A B)))))))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = fma(B, B, (C * (-4.0 * A)));
	double t_1 = (B * B) - ((A * C) * 4.0);
	double tmp;
	if (B <= -7.6e+78) {
		tmp = sqrt(2.0) * -sqrt((F / B));
	} else if (B <= -1.32e-55) {
		tmp = -sqrt((2.0 * ((F * t_0) * (A - (hypot(B, (A - C)) - C))))) / t_0;
	} else if (B <= 2.5e-96) {
		tmp = -sqrt((2.0 * ((F * t_1) * (2.0 * A)))) / t_1;
	} else {
		tmp = (sqrt(2.0) / B) * -sqrt((F * (A - hypot(A, B))));
	}
	return tmp;
}

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = fma(B, B, Float64(C * Float64(-4.0 * A)))
	t_1 = Float64(Float64(B * B) - Float64(Float64(A * C) * 4.0))
	tmp = 0.0
	if (B <= -7.6e+78)
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F / B))));
	elseif (B <= -1.32e-55)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(F * t_0) * Float64(A - Float64(hypot(B, Float64(A - C)) - C)))))) / t_0);
	elseif (B <= 2.5e-96)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(F * t_1) * Float64(2.0 * A))))) / t_1);
	else
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(-sqrt(Float64(F * Float64(A - hypot(A, B))))));
	end
	return tmp
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(B * B + N[(C * N[(-4.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(B * B), $MachinePrecision] - N[(N[(A * C), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -7.6e+78], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[B, -1.32e-55], N[((-N[Sqrt[N[(2.0 * N[(N[(F * t$95$0), $MachinePrecision] * N[(A - N[(N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision] - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[B, 2.5e-96], N[((-N[Sqrt[N[(2.0 * N[(N[(F * t$95$1), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * (-N[Sqrt[N[(F * N[(A - N[Sqrt[A ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < -7.5999999999999998e78

   1. Initial program 14.3%

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

   3. Simplified 14.3%

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

   4. Taylor expanded in C around 0 14.4%

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

      1. +-commutative 14.4%

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

      2. unpow2 14.4%

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

      3. unpow2 14.4%

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

      4. hypot-def 14.4%

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

   6. Simplified 14.4%

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

   7. Taylor expanded in B around -inf 6.6%

      \[\leadsto \frac{-\sqrt{2 \cdot \color{blue}{\left(F \cdot {B}^{3}\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   8. Taylor expanded in A around 0 49.6%

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

      1. mul-1-neg 49.6%

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

      2. distribute-rgt-neg-in 49.6%

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

   10. Simplified 49.6%

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

   if -7.5999999999999998e78 < B < -1.31999999999999993e-55

   1. Initial program 33.5%

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

   3. Simplified 40.8%

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

   if -1.31999999999999993e-55 < B < 2.49999999999999997e-96

   1. Initial program 15.1%

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

   3. Simplified 15.1%

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

   4. Taylor expanded in A around -inf 22.3%

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

   if 2.49999999999999997e-96 < B

   1. Initial program 21.2%

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

   3. Simplified 21.2%

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

   4. Taylor expanded in C around 0 22.7%

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

      1. mul-1-neg 22.7%

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

      2. +-commutative 22.7%

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

      3. unpow2 22.7%

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

      4. unpow2 22.7%

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

      5. hypot-def 49.4%

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

   6. Simplified 49.4%

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

4. Final simplification 39.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -7.6 \cdot 10^{+78}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;B \leq -1.32 \cdot 10^{-55}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)\right) \cdot \left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)}\\ \mathbf{elif}\;B \leq 2.5 \cdot 10^{-96}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - \left(A \cdot C\right) \cdot 4\right)\right) \cdot \left(2 \cdot A\right)\right)}}{B \cdot B - \left(A \cdot C\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}\right)\\ \end{array} \]

Alternative 3: 45.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - \left(A \cdot C\right) \cdot 4\\ \mathbf{if}\;B \leq -2 \cdot 10^{+79}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;B \leq -1.5 \cdot 10^{-55}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \left(-\sqrt{\left(-4 \cdot \left(A \cdot C\right) + B \cdot B\right) \cdot \left(F \cdot \left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right)\right)}\right)}{t_0}\\ \mathbf{elif}\;B \leq 2.5 \cdot 10^{-96}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(2 \cdot A\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* (* A C) 4.0))))
   (if (<= B -2e+79)
     (* (sqrt 2.0) (- (sqrt (/ F B))))
     (if (<= B -1.5e-55)
       (/
        (*
         (sqrt 2.0)
         (-
          (sqrt
           (*
            (+ (* -4.0 (* A C)) (* B B))
            (* F (- A (- (hypot B (- A C)) C)))))))
        t_0)
       (if (<= B 2.5e-96)
         (/ (- (sqrt (* 2.0 (* (* F t_0) (* 2.0 A))))) t_0)
         (* (/ (sqrt 2.0) B) (- (sqrt (* F (- A (hypot A B)))))))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - ((A * C) * 4.0);
	double tmp;
	if (B <= -2e+79) {
		tmp = sqrt(2.0) * -sqrt((F / B));
	} else if (B <= -1.5e-55) {
		tmp = (sqrt(2.0) * -sqrt((((-4.0 * (A * C)) + (B * B)) * (F * (A - (hypot(B, (A - C)) - C)))))) / t_0;
	} else if (B <= 2.5e-96) {
		tmp = -sqrt((2.0 * ((F * t_0) * (2.0 * A)))) / t_0;
	} else {
		tmp = (sqrt(2.0) / B) * -sqrt((F * (A - hypot(A, B))));
	}
	return tmp;
}

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - ((A * C) * 4.0);
	double tmp;
	if (B <= -2e+79) {
		tmp = Math.sqrt(2.0) * -Math.sqrt((F / B));
	} else if (B <= -1.5e-55) {
		tmp = (Math.sqrt(2.0) * -Math.sqrt((((-4.0 * (A * C)) + (B * B)) * (F * (A - (Math.hypot(B, (A - C)) - C)))))) / t_0;
	} else if (B <= 2.5e-96) {
		tmp = -Math.sqrt((2.0 * ((F * t_0) * (2.0 * A)))) / t_0;
	} else {
		tmp = (Math.sqrt(2.0) / B) * -Math.sqrt((F * (A - Math.hypot(A, B))));
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (B * B) - ((A * C) * 4.0)
	tmp = 0
	if B <= -2e+79:
		tmp = math.sqrt(2.0) * -math.sqrt((F / B))
	elif B <= -1.5e-55:
		tmp = (math.sqrt(2.0) * -math.sqrt((((-4.0 * (A * C)) + (B * B)) * (F * (A - (math.hypot(B, (A - C)) - C)))))) / t_0
	elif B <= 2.5e-96:
		tmp = -math.sqrt((2.0 * ((F * t_0) * (2.0 * A)))) / t_0
	else:
		tmp = (math.sqrt(2.0) / B) * -math.sqrt((F * (A - math.hypot(A, B))))
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(Float64(A * C) * 4.0))
	tmp = 0.0
	if (B <= -2e+79)
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F / B))));
	elseif (B <= -1.5e-55)
		tmp = Float64(Float64(sqrt(2.0) * Float64(-sqrt(Float64(Float64(Float64(-4.0 * Float64(A * C)) + Float64(B * B)) * Float64(F * Float64(A - Float64(hypot(B, Float64(A - C)) - C))))))) / t_0);
	elseif (B <= 2.5e-96)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(F * t_0) * Float64(2.0 * A))))) / t_0);
	else
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(-sqrt(Float64(F * Float64(A - hypot(A, B))))));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - ((A * C) * 4.0);
	tmp = 0.0;
	if (B <= -2e+79)
		tmp = sqrt(2.0) * -sqrt((F / B));
	elseif (B <= -1.5e-55)
		tmp = (sqrt(2.0) * -sqrt((((-4.0 * (A * C)) + (B * B)) * (F * (A - (hypot(B, (A - C)) - C)))))) / t_0;
	elseif (B <= 2.5e-96)
		tmp = -sqrt((2.0 * ((F * t_0) * (2.0 * A)))) / t_0;
	else
		tmp = (sqrt(2.0) / B) * -sqrt((F * (A - hypot(A, B))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(N[(A * C), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -2e+79], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[B, -1.5e-55], N[(N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(N[(N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision] * N[(F * N[(A - N[(N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision] - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[B, 2.5e-96], N[((-N[Sqrt[N[(2.0 * N[(N[(F * t$95$0), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * (-N[Sqrt[N[(F * N[(A - N[Sqrt[A ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if B < -1.99999999999999993e79

   1. Initial program 14.3%

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

   3. Simplified 14.3%

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

   4. Taylor expanded in C around 0 14.4%

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

      1. +-commutative 14.4%

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

      2. unpow2 14.4%

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

      3. unpow2 14.4%

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

      4. hypot-def 14.4%

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

   6. Simplified 14.4%

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

   7. Taylor expanded in B around -inf 6.6%

      \[\leadsto \frac{-\sqrt{2 \cdot \color{blue}{\left(F \cdot {B}^{3}\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   8. Taylor expanded in A around 0 49.6%

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

      1. mul-1-neg 49.6%

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

      2. distribute-rgt-neg-in 49.6%

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

   10. Simplified 49.6%

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

   if -1.99999999999999993e79 < B < -1.50000000000000008e-55

   1. Initial program 33.5%

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

   3. Simplified 33.5%

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

      1. sqrt-prod 33.5%

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

      2. associate-*l* 33.4%

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

      3. cancel-sign-sub-inv 33.4%

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

      4. metadata-eval 33.4%

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

      5. associate--l+ 33.4%

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

      6. unpow2 33.4%

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

      7. hypot-udef 40.9%

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

   5. Applied egg-rr 40.9%

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

   if -1.50000000000000008e-55 < B < 2.49999999999999997e-96

   1. Initial program 15.1%

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

   3. Simplified 15.1%

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

   4. Taylor expanded in A around -inf 22.3%

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

   if 2.49999999999999997e-96 < B

   1. Initial program 21.2%

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

   3. Simplified 21.2%

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

   4. Taylor expanded in C around 0 22.7%

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

      1. mul-1-neg 22.7%

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

      2. +-commutative 22.7%

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

      3. unpow2 22.7%

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

      4. unpow2 22.7%

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

      5. hypot-def 49.4%

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

   6. Simplified 49.4%

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

4. Final simplification 39.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -2 \cdot 10^{+79}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;B \leq -1.5 \cdot 10^{-55}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \left(-\sqrt{\left(-4 \cdot \left(A \cdot C\right) + B \cdot B\right) \cdot \left(F \cdot \left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right)\right)}\right)}{B \cdot B - \left(A \cdot C\right) \cdot 4}\\ \mathbf{elif}\;B \leq 2.5 \cdot 10^{-96}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - \left(A \cdot C\right) \cdot 4\right)\right) \cdot \left(2 \cdot A\right)\right)}}{B \cdot B - \left(A \cdot C\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}\right)\\ \end{array} \]

Alternative 4: 45.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right) + B \cdot B\\ t_1 := B \cdot B - \left(A \cdot C\right) \cdot 4\\ \mathbf{if}\;B \leq -1.85 \cdot 10^{+79}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;B \leq -2.1 \cdot 10^{-55}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 2.5 \cdot 10^{-96}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_1\right) \cdot \left(2 \cdot A\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (+ (* -4.0 (* A C)) (* B B))) (t_1 (- (* B B) (* (* A C) 4.0))))
   (if (<= B -1.85e+79)
     (* (sqrt 2.0) (- (sqrt (/ F B))))
     (if (<= B -2.1e-55)
       (/ (- (sqrt (* 2.0 (* t_0 (* F (- A (- (hypot B (- A C)) C))))))) t_0)
       (if (<= B 2.5e-96)
         (/ (- (sqrt (* 2.0 (* (* F t_1) (* 2.0 A))))) t_1)
         (* (/ (sqrt 2.0) B) (- (sqrt (* F (- A (hypot A B)))))))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (-4.0 * (A * C)) + (B * B);
	double t_1 = (B * B) - ((A * C) * 4.0);
	double tmp;
	if (B <= -1.85e+79) {
		tmp = sqrt(2.0) * -sqrt((F / B));
	} else if (B <= -2.1e-55) {
		tmp = -sqrt((2.0 * (t_0 * (F * (A - (hypot(B, (A - C)) - C)))))) / t_0;
	} else if (B <= 2.5e-96) {
		tmp = -sqrt((2.0 * ((F * t_1) * (2.0 * A)))) / t_1;
	} else {
		tmp = (sqrt(2.0) / B) * -sqrt((F * (A - hypot(A, B))));
	}
	return tmp;
}

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = (-4.0 * (A * C)) + (B * B);
	double t_1 = (B * B) - ((A * C) * 4.0);
	double tmp;
	if (B <= -1.85e+79) {
		tmp = Math.sqrt(2.0) * -Math.sqrt((F / B));
	} else if (B <= -2.1e-55) {
		tmp = -Math.sqrt((2.0 * (t_0 * (F * (A - (Math.hypot(B, (A - C)) - C)))))) / t_0;
	} else if (B <= 2.5e-96) {
		tmp = -Math.sqrt((2.0 * ((F * t_1) * (2.0 * A)))) / t_1;
	} else {
		tmp = (Math.sqrt(2.0) / B) * -Math.sqrt((F * (A - Math.hypot(A, B))));
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (-4.0 * (A * C)) + (B * B)
	t_1 = (B * B) - ((A * C) * 4.0)
	tmp = 0
	if B <= -1.85e+79:
		tmp = math.sqrt(2.0) * -math.sqrt((F / B))
	elif B <= -2.1e-55:
		tmp = -math.sqrt((2.0 * (t_0 * (F * (A - (math.hypot(B, (A - C)) - C)))))) / t_0
	elif B <= 2.5e-96:
		tmp = -math.sqrt((2.0 * ((F * t_1) * (2.0 * A)))) / t_1
	else:
		tmp = (math.sqrt(2.0) / B) * -math.sqrt((F * (A - math.hypot(A, B))))
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(-4.0 * Float64(A * C)) + Float64(B * B))
	t_1 = Float64(Float64(B * B) - Float64(Float64(A * C) * 4.0))
	tmp = 0.0
	if (B <= -1.85e+79)
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F / B))));
	elseif (B <= -2.1e-55)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_0 * Float64(F * Float64(A - Float64(hypot(B, Float64(A - C)) - C))))))) / t_0);
	elseif (B <= 2.5e-96)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(F * t_1) * Float64(2.0 * A))))) / t_1);
	else
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(-sqrt(Float64(F * Float64(A - hypot(A, B))))));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (-4.0 * (A * C)) + (B * B);
	t_1 = (B * B) - ((A * C) * 4.0);
	tmp = 0.0;
	if (B <= -1.85e+79)
		tmp = sqrt(2.0) * -sqrt((F / B));
	elseif (B <= -2.1e-55)
		tmp = -sqrt((2.0 * (t_0 * (F * (A - (hypot(B, (A - C)) - C)))))) / t_0;
	elseif (B <= 2.5e-96)
		tmp = -sqrt((2.0 * ((F * t_1) * (2.0 * A)))) / t_1;
	else
		tmp = (sqrt(2.0) / B) * -sqrt((F * (A - hypot(A, B))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(B * B), $MachinePrecision] - N[(N[(A * C), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -1.85e+79], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[B, -2.1e-55], N[((-N[Sqrt[N[(2.0 * N[(t$95$0 * N[(F * N[(A - N[(N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision] - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[B, 2.5e-96], N[((-N[Sqrt[N[(2.0 * N[(N[(F * t$95$1), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * (-N[Sqrt[N[(F * N[(A - N[Sqrt[A ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < -1.85000000000000005e79

   1. Initial program 14.3%

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

   3. Simplified 14.3%

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

   4. Taylor expanded in C around 0 14.4%

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

      1. +-commutative 14.4%

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

      2. unpow2 14.4%

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

      3. unpow2 14.4%

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

      4. hypot-def 14.4%

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

   6. Simplified 14.4%

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

   7. Taylor expanded in B around -inf 6.6%

      \[\leadsto \frac{-\sqrt{2 \cdot \color{blue}{\left(F \cdot {B}^{3}\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   8. Taylor expanded in A around 0 49.6%

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

      1. mul-1-neg 49.6%

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

      2. distribute-rgt-neg-in 49.6%

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

   10. Simplified 49.6%

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

   if -1.85000000000000005e79 < B < -2.1000000000000002e-55

   1. Initial program 33.5%

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

   3. Simplified 33.5%

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

      1. distribute-frac-neg 33.5%

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

   5. Applied egg-rr 40.9%

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

   if -2.1000000000000002e-55 < B < 2.49999999999999997e-96

   1. Initial program 15.1%

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

   3. Simplified 15.1%

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

   4. Taylor expanded in A around -inf 22.3%

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

   if 2.49999999999999997e-96 < B

   1. Initial program 21.2%

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

   3. Simplified 21.2%

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

   4. Taylor expanded in C around 0 22.7%

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

      1. mul-1-neg 22.7%

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

      2. +-commutative 22.7%

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

      3. unpow2 22.7%

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

      4. unpow2 22.7%

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

      5. hypot-def 49.4%

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

   6. Simplified 49.4%

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

4. Final simplification 39.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -1.85 \cdot 10^{+79}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;B \leq -2.1 \cdot 10^{-55}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(-4 \cdot \left(A \cdot C\right) + B \cdot B\right) \cdot \left(F \cdot \left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right)\right)\right)}}{-4 \cdot \left(A \cdot C\right) + B \cdot B}\\ \mathbf{elif}\;B \leq 2.5 \cdot 10^{-96}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - \left(A \cdot C\right) \cdot 4\right)\right) \cdot \left(2 \cdot A\right)\right)}}{B \cdot B - \left(A \cdot C\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}\right)\\ \end{array} \]

Alternative 5: 35.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - \left(A \cdot C\right) \cdot 4\\ t_1 := F \cdot t_0\\ t_2 := A \cdot A - A \cdot A\\ \mathbf{if}\;C \leq 6.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot t_1\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(A + -0.5 \cdot \left(\frac{B \cdot B + t_2}{C} - \frac{A}{C} \cdot \frac{t_2 - B \cdot B}{C}\right)\right)\right)\right)}}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* (* A C) 4.0)))
        (t_1 (* F t_0))
        (t_2 (- (* A A) (* A A))))
   (if (<= C 6.6e-9)
     (/ (- (sqrt (* 2.0 (* (- A (hypot A B)) t_1)))) t_0)
     (-
      (/
       (sqrt
        (*
         2.0
         (*
          t_1
          (+
           A
           (+
            A
            (*
             -0.5
             (- (/ (+ (* B B) t_2) C) (* (/ A C) (/ (- t_2 (* B B)) C)))))))))
       t_0)))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - ((A * C) * 4.0);
	double t_1 = F * t_0;
	double t_2 = (A * A) - (A * A);
	double tmp;
	if (C <= 6.6e-9) {
		tmp = -sqrt((2.0 * ((A - hypot(A, B)) * t_1))) / t_0;
	} else {
		tmp = -(sqrt((2.0 * (t_1 * (A + (A + (-0.5 * ((((B * B) + t_2) / C) - ((A / C) * ((t_2 - (B * B)) / C))))))))) / t_0);
	}
	return tmp;
}

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - ((A * C) * 4.0);
	double t_1 = F * t_0;
	double t_2 = (A * A) - (A * A);
	double tmp;
	if (C <= 6.6e-9) {
		tmp = -Math.sqrt((2.0 * ((A - Math.hypot(A, B)) * t_1))) / t_0;
	} else {
		tmp = -(Math.sqrt((2.0 * (t_1 * (A + (A + (-0.5 * ((((B * B) + t_2) / C) - ((A / C) * ((t_2 - (B * B)) / C))))))))) / t_0);
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (B * B) - ((A * C) * 4.0)
	t_1 = F * t_0
	t_2 = (A * A) - (A * A)
	tmp = 0
	if C <= 6.6e-9:
		tmp = -math.sqrt((2.0 * ((A - math.hypot(A, B)) * t_1))) / t_0
	else:
		tmp = -(math.sqrt((2.0 * (t_1 * (A + (A + (-0.5 * ((((B * B) + t_2) / C) - ((A / C) * ((t_2 - (B * B)) / C))))))))) / t_0)
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(Float64(A * C) * 4.0))
	t_1 = Float64(F * t_0)
	t_2 = Float64(Float64(A * A) - Float64(A * A))
	tmp = 0.0
	if (C <= 6.6e-9)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(A - hypot(A, B)) * t_1)))) / t_0);
	else
		tmp = Float64(-Float64(sqrt(Float64(2.0 * Float64(t_1 * Float64(A + Float64(A + Float64(-0.5 * Float64(Float64(Float64(Float64(B * B) + t_2) / C) - Float64(Float64(A / C) * Float64(Float64(t_2 - Float64(B * B)) / C))))))))) / t_0));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - ((A * C) * 4.0);
	t_1 = F * t_0;
	t_2 = (A * A) - (A * A);
	tmp = 0.0;
	if (C <= 6.6e-9)
		tmp = -sqrt((2.0 * ((A - hypot(A, B)) * t_1))) / t_0;
	else
		tmp = -(sqrt((2.0 * (t_1 * (A + (A + (-0.5 * ((((B * B) + t_2) / C) - ((A / C) * ((t_2 - (B * B)) / C))))))))) / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(N[(A * C), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(F * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(A * A), $MachinePrecision] - N[(A * A), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[C, 6.6e-9], N[((-N[Sqrt[N[(2.0 * N[(N[(A - N[Sqrt[A ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], (-N[(N[Sqrt[N[(2.0 * N[(t$95$1 * N[(A + N[(A + N[(-0.5 * N[(N[(N[(N[(B * B), $MachinePrecision] + t$95$2), $MachinePrecision] / C), $MachinePrecision] - N[(N[(A / C), $MachinePrecision] * N[(N[(t$95$2 - N[(B * B), $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision])]]]]

Derivation

1. Split input into 2 regimes

2. if C < 6.60000000000000037e-9

   1. Initial program 25.2%

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

   3. Simplified 25.2%

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

   4. Taylor expanded in C around 0 22.3%

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

      1. +-commutative 22.3%

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

      2. unpow2 22.3%

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

      3. unpow2 22.3%

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

      4. hypot-def 25.5%

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

   6. Simplified 25.5%

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

   if 6.60000000000000037e-9 < C

   1. Initial program 1.1%

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

   3. Simplified 1.1%

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

   4. Taylor expanded in C around inf 29.1%

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

      1. associate--l+ 29.1%

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

   6. Simplified 29.1%

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

4. Final simplification 26.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 6.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot \left(F \cdot \left(B \cdot B - \left(A \cdot C\right) \cdot 4\right)\right)\right)}}{B \cdot B - \left(A \cdot C\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - \left(A \cdot C\right) \cdot 4\right)\right) \cdot \left(A + \left(A + -0.5 \cdot \left(\frac{B \cdot B + \left(A \cdot A - A \cdot A\right)}{C} - \frac{A}{C} \cdot \frac{\left(A \cdot A - A \cdot A\right) - B \cdot B}{C}\right)\right)\right)\right)}}{B \cdot B - \left(A \cdot C\right) \cdot 4}\\ \end{array} \]

Alternative 6: 36.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - \left(A \cdot C\right) \cdot 4\\ \mathbf{if}\;B \leq -5.5 \cdot 10^{-10}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;B \leq 2.5 \cdot 10^{-96}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(2 \cdot A\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 3.5 \cdot 10^{+149}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot \left(F \cdot \left(B \cdot B\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{F \cdot A}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* (* A C) 4.0))))
   (if (<= B -5.5e-10)
     (* (sqrt 2.0) (- (sqrt (/ F B))))
     (if (<= B 2.5e-96)
       (/ (- (sqrt (* 2.0 (* (* F t_0) (* 2.0 A))))) t_0)
       (if (<= B 3.5e+149)
         (/ (- (sqrt (* 2.0 (* (- A (hypot B A)) (* F (* B B)))))) t_0)
         (* -2.0 (/ (sqrt (* F A)) B)))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - ((A * C) * 4.0);
	double tmp;
	if (B <= -5.5e-10) {
		tmp = sqrt(2.0) * -sqrt((F / B));
	} else if (B <= 2.5e-96) {
		tmp = -sqrt((2.0 * ((F * t_0) * (2.0 * A)))) / t_0;
	} else if (B <= 3.5e+149) {
		tmp = -sqrt((2.0 * ((A - hypot(B, A)) * (F * (B * B))))) / t_0;
	} else {
		tmp = -2.0 * (sqrt((F * A)) / B);
	}
	return tmp;
}

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - ((A * C) * 4.0);
	double tmp;
	if (B <= -5.5e-10) {
		tmp = Math.sqrt(2.0) * -Math.sqrt((F / B));
	} else if (B <= 2.5e-96) {
		tmp = -Math.sqrt((2.0 * ((F * t_0) * (2.0 * A)))) / t_0;
	} else if (B <= 3.5e+149) {
		tmp = -Math.sqrt((2.0 * ((A - Math.hypot(B, A)) * (F * (B * B))))) / t_0;
	} else {
		tmp = -2.0 * (Math.sqrt((F * A)) / B);
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (B * B) - ((A * C) * 4.0)
	tmp = 0
	if B <= -5.5e-10:
		tmp = math.sqrt(2.0) * -math.sqrt((F / B))
	elif B <= 2.5e-96:
		tmp = -math.sqrt((2.0 * ((F * t_0) * (2.0 * A)))) / t_0
	elif B <= 3.5e+149:
		tmp = -math.sqrt((2.0 * ((A - math.hypot(B, A)) * (F * (B * B))))) / t_0
	else:
		tmp = -2.0 * (math.sqrt((F * A)) / B)
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(Float64(A * C) * 4.0))
	tmp = 0.0
	if (B <= -5.5e-10)
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F / B))));
	elseif (B <= 2.5e-96)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(F * t_0) * Float64(2.0 * A))))) / t_0);
	elseif (B <= 3.5e+149)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(A - hypot(B, A)) * Float64(F * Float64(B * B)))))) / t_0);
	else
		tmp = Float64(-2.0 * Float64(sqrt(Float64(F * A)) / B));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - ((A * C) * 4.0);
	tmp = 0.0;
	if (B <= -5.5e-10)
		tmp = sqrt(2.0) * -sqrt((F / B));
	elseif (B <= 2.5e-96)
		tmp = -sqrt((2.0 * ((F * t_0) * (2.0 * A)))) / t_0;
	elseif (B <= 3.5e+149)
		tmp = -sqrt((2.0 * ((A - hypot(B, A)) * (F * (B * B))))) / t_0;
	else
		tmp = -2.0 * (sqrt((F * A)) / B);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(N[(A * C), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -5.5e-10], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[B, 2.5e-96], N[((-N[Sqrt[N[(2.0 * N[(N[(F * t$95$0), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[B, 3.5e+149], N[((-N[Sqrt[N[(2.0 * N[(N[(A - N[Sqrt[B ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision] * N[(F * N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(-2.0 * N[(N[Sqrt[N[(F * A), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if B < -5.4999999999999996e-10

   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. Step-by-step derivation

   3. Simplified 19.6%

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

   4. Taylor expanded in C around 0 19.8%

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

      1. +-commutative 19.8%

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

      2. unpow2 19.8%

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

      3. unpow2 19.8%

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

      4. hypot-def 21.0%

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

   6. Simplified 21.0%

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

   7. Taylor expanded in B around -inf 14.1%

      \[\leadsto \frac{-\sqrt{2 \cdot \color{blue}{\left(F \cdot {B}^{3}\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   8. Taylor expanded in A around 0 41.6%

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

      1. mul-1-neg 41.6%

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

      2. distribute-rgt-neg-in 41.6%

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

   10. Simplified 41.6%

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

   if -5.4999999999999996e-10 < B < 2.49999999999999997e-96

   1. Initial program 18.1%

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

   3. Simplified 18.1%

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

   4. Taylor expanded in A around -inf 21.8%

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

   if 2.49999999999999997e-96 < B < 3.50000000000000011e149

   1. Initial program 39.1%

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

   3. Simplified 39.1%

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

   4. Taylor expanded in C around 0 36.3%

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

      1. +-commutative 36.3%

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

      2. unpow2 36.3%

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

      3. unpow2 36.3%

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

      4. hypot-def 38.5%

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

   6. Simplified 38.5%

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

   7. Taylor expanded in C around 0 36.7%

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

      1. unpow2 36.7%

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

      2. unpow2 36.7%

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

      3. hypot-def 37.0%

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

      4. unpow2 37.0%

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

   9. Simplified 37.0%

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

   if 3.50000000000000011e149 < B

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in A around -inf 0.0%

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

   5. Taylor expanded in B around inf 6.9%

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

      1. un-div-inv 6.9%

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

   7. Applied egg-rr 6.9%

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

4. Final simplification 28.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -5.5 \cdot 10^{-10}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;B \leq 2.5 \cdot 10^{-96}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - \left(A \cdot C\right) \cdot 4\right)\right) \cdot \left(2 \cdot A\right)\right)}}{B \cdot B - \left(A \cdot C\right) \cdot 4}\\ \mathbf{elif}\;B \leq 3.5 \cdot 10^{+149}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot \left(F \cdot \left(B \cdot B\right)\right)\right)}}{B \cdot B - \left(A \cdot C\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{F \cdot A}}{B}\\ \end{array} \]

Alternative 7: 36.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - \left(A \cdot C\right) \cdot 4\\ \mathbf{if}\;B \leq -1.9 \cdot 10^{-9}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;B \leq 7.5 \cdot 10^{-65}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(2 \cdot A\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 4.9 \cdot 10^{+148}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(B \cdot F\right) \cdot \left(-B \cdot B\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{F \cdot A}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* (* A C) 4.0))))
   (if (<= B -1.9e-9)
     (* (sqrt 2.0) (- (sqrt (/ F B))))
     (if (<= B 7.5e-65)
       (/ (- (sqrt (* 2.0 (* (* F t_0) (* 2.0 A))))) t_0)
       (if (<= B 4.9e+148)
         (/
          (- (sqrt (* 2.0 (* (* B F) (- (* B B))))))
          (fma B B (* A (* -4.0 C))))
         (* -2.0 (/ (sqrt (* F A)) B)))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - ((A * C) * 4.0);
	double tmp;
	if (B <= -1.9e-9) {
		tmp = sqrt(2.0) * -sqrt((F / B));
	} else if (B <= 7.5e-65) {
		tmp = -sqrt((2.0 * ((F * t_0) * (2.0 * A)))) / t_0;
	} else if (B <= 4.9e+148) {
		tmp = -sqrt((2.0 * ((B * F) * -(B * B)))) / fma(B, B, (A * (-4.0 * C)));
	} else {
		tmp = -2.0 * (sqrt((F * A)) / B);
	}
	return tmp;
}

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(Float64(A * C) * 4.0))
	tmp = 0.0
	if (B <= -1.9e-9)
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F / B))));
	elseif (B <= 7.5e-65)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(F * t_0) * Float64(2.0 * A))))) / t_0);
	elseif (B <= 4.9e+148)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(B * F) * Float64(-Float64(B * B)))))) / fma(B, B, Float64(A * Float64(-4.0 * C))));
	else
		tmp = Float64(-2.0 * Float64(sqrt(Float64(F * A)) / B));
	end
	return tmp
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(N[(A * C), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -1.9e-9], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[B, 7.5e-65], N[((-N[Sqrt[N[(2.0 * N[(N[(F * t$95$0), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[B, 4.9e+148], N[((-N[Sqrt[N[(2.0 * N[(N[(B * F), $MachinePrecision] * (-N[(B * B), $MachinePrecision])), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(B * B + N[(A * N[(-4.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[Sqrt[N[(F * A), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if B < -1.90000000000000006e-9

   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. Step-by-step derivation

   3. Simplified 19.6%

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

   4. Taylor expanded in C around 0 19.8%

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

      1. +-commutative 19.8%

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

      2. unpow2 19.8%

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

      3. unpow2 19.8%

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

      4. hypot-def 21.0%

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

   6. Simplified 21.0%

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

   7. Taylor expanded in B around -inf 14.1%

      \[\leadsto \frac{-\sqrt{2 \cdot \color{blue}{\left(F \cdot {B}^{3}\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   8. Taylor expanded in A around 0 41.6%

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

      1. mul-1-neg 41.6%

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

      2. distribute-rgt-neg-in 41.6%

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

   10. Simplified 41.6%

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

   if -1.90000000000000006e-9 < B < 7.5000000000000002e-65

   1. Initial program 20.1%

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

   3. Simplified 20.1%

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

   4. Taylor expanded in A around -inf 21.6%

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

   if 7.5000000000000002e-65 < B < 4.9e148

   1. Initial program 37.3%

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

   3. Simplified 33.4%

      \[\leadsto \color{blue}{\frac{-\sqrt{F \cdot \left(\left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right) \cdot \mathsf{fma}\left(C, A \cdot -8, 2 \cdot \left(B \cdot B\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]

   4. Taylor expanded in C around 0 37.9%

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

      1. associate-*r* 38.0%

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

      2. +-commutative 38.0%

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

      3. unpow2 38.0%

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

      4. unpow2 38.0%

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

      5. hypot-def 38.2%

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

      6. unpow2 38.2%

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

   6. Simplified 38.2%

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

   7. Taylor expanded in A around 0 38.7%

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

      1. mul-1-neg 38.7%

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

   9. Simplified 38.7%

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

   if 4.9e148 < B

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in A around -inf 0.0%

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

   5. Taylor expanded in B around inf 6.9%

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

      1. un-div-inv 6.9%

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

   7. Applied egg-rr 6.9%

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

4. Final simplification 28.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -1.9 \cdot 10^{-9}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;B \leq 7.5 \cdot 10^{-65}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - \left(A \cdot C\right) \cdot 4\right)\right) \cdot \left(2 \cdot A\right)\right)}}{B \cdot B - \left(A \cdot C\right) \cdot 4}\\ \mathbf{elif}\;B \leq 4.9 \cdot 10^{+148}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(B \cdot F\right) \cdot \left(-B \cdot B\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{F \cdot A}}{B}\\ \end{array} \]

Alternative 8: 36.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := \left(A \cdot C\right) \cdot 4\\ t_1 := B \cdot B - t_0\\ \mathbf{if}\;B \leq -1.5 \cdot 10^{-9}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;B \leq 3.8 \cdot 10^{-65}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_1\right) \cdot \left(2 \cdot A\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(B \cdot \left(F \cdot \left(t_0 - B \cdot B\right)\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{F \cdot A}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (* (* A C) 4.0)) (t_1 (- (* B B) t_0)))
   (if (<= B -1.5e-9)
     (* (sqrt 2.0) (- (sqrt (/ F B))))
     (if (<= B 3.8e-65)
       (/ (- (sqrt (* 2.0 (* (* F t_1) (* 2.0 A))))) t_1)
       (if (<= B 1.32e+154)
         (/ (- (sqrt (* 2.0 (* B (* F (- t_0 (* B B))))))) t_1)
         (* -2.0 (/ (sqrt (* F A)) B)))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (A * C) * 4.0;
	double t_1 = (B * B) - t_0;
	double tmp;
	if (B <= -1.5e-9) {
		tmp = sqrt(2.0) * -sqrt((F / B));
	} else if (B <= 3.8e-65) {
		tmp = -sqrt((2.0 * ((F * t_1) * (2.0 * A)))) / t_1;
	} else if (B <= 1.32e+154) {
		tmp = -sqrt((2.0 * (B * (F * (t_0 - (B * B)))))) / t_1;
	} else {
		tmp = -2.0 * (sqrt((F * A)) / B);
	}
	return tmp;
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this function.
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
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (a * c) * 4.0d0
    t_1 = (b * b) - t_0
    if (b <= (-1.5d-9)) then
        tmp = sqrt(2.0d0) * -sqrt((f / b))
    else if (b <= 3.8d-65) then
        tmp = -sqrt((2.0d0 * ((f * t_1) * (2.0d0 * a)))) / t_1
    else if (b <= 1.32d+154) then
        tmp = -sqrt((2.0d0 * (b * (f * (t_0 - (b * b)))))) / t_1
    else
        tmp = (-2.0d0) * (sqrt((f * a)) / b)
    end if
    code = tmp
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = (A * C) * 4.0;
	double t_1 = (B * B) - t_0;
	double tmp;
	if (B <= -1.5e-9) {
		tmp = Math.sqrt(2.0) * -Math.sqrt((F / B));
	} else if (B <= 3.8e-65) {
		tmp = -Math.sqrt((2.0 * ((F * t_1) * (2.0 * A)))) / t_1;
	} else if (B <= 1.32e+154) {
		tmp = -Math.sqrt((2.0 * (B * (F * (t_0 - (B * B)))))) / t_1;
	} else {
		tmp = -2.0 * (Math.sqrt((F * A)) / B);
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (A * C) * 4.0
	t_1 = (B * B) - t_0
	tmp = 0
	if B <= -1.5e-9:
		tmp = math.sqrt(2.0) * -math.sqrt((F / B))
	elif B <= 3.8e-65:
		tmp = -math.sqrt((2.0 * ((F * t_1) * (2.0 * A)))) / t_1
	elif B <= 1.32e+154:
		tmp = -math.sqrt((2.0 * (B * (F * (t_0 - (B * B)))))) / t_1
	else:
		tmp = -2.0 * (math.sqrt((F * A)) / B)
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(A * C) * 4.0)
	t_1 = Float64(Float64(B * B) - t_0)
	tmp = 0.0
	if (B <= -1.5e-9)
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F / B))));
	elseif (B <= 3.8e-65)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(F * t_1) * Float64(2.0 * A))))) / t_1);
	elseif (B <= 1.32e+154)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(B * Float64(F * Float64(t_0 - Float64(B * B))))))) / t_1);
	else
		tmp = Float64(-2.0 * Float64(sqrt(Float64(F * A)) / B));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (A * C) * 4.0;
	t_1 = (B * B) - t_0;
	tmp = 0.0;
	if (B <= -1.5e-9)
		tmp = sqrt(2.0) * -sqrt((F / B));
	elseif (B <= 3.8e-65)
		tmp = -sqrt((2.0 * ((F * t_1) * (2.0 * A)))) / t_1;
	elseif (B <= 1.32e+154)
		tmp = -sqrt((2.0 * (B * (F * (t_0 - (B * B)))))) / t_1;
	else
		tmp = -2.0 * (sqrt((F * A)) / B);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(A * C), $MachinePrecision] * 4.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(B * B), $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[B, -1.5e-9], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[B, 3.8e-65], N[((-N[Sqrt[N[(2.0 * N[(N[(F * t$95$1), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], If[LessEqual[B, 1.32e+154], N[((-N[Sqrt[N[(2.0 * N[(B * N[(F * N[(t$95$0 - N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], N[(-2.0 * N[(N[Sqrt[N[(F * A), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < -1.49999999999999999e-9

   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. Step-by-step derivation

   3. Simplified 19.6%

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

   4. Taylor expanded in C around 0 19.8%

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

      1. +-commutative 19.8%

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

      2. unpow2 19.8%

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

      3. unpow2 19.8%

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

      4. hypot-def 21.0%

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

   6. Simplified 21.0%

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

   7. Taylor expanded in B around -inf 14.1%

      \[\leadsto \frac{-\sqrt{2 \cdot \color{blue}{\left(F \cdot {B}^{3}\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   8. Taylor expanded in A around 0 41.6%

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

      1. mul-1-neg 41.6%

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

      2. distribute-rgt-neg-in 41.6%

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

   10. Simplified 41.6%

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

   if -1.49999999999999999e-9 < B < 3.8000000000000002e-65

   1. Initial program 20.1%

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

   3. Simplified 20.1%

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

   4. Taylor expanded in A around -inf 21.6%

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

   if 3.8000000000000002e-65 < B < 1.31999999999999998e154

   1. Initial program 37.3%

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

   3. Simplified 37.3%

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

   4. Taylor expanded in A around 0 37.7%

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

      1. +-commutative 37.7%

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

      2. unpow2 37.7%

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

      3. unpow2 37.7%

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

      4. hypot-def 38.0%

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

   6. Simplified 38.0%

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

   7. Taylor expanded in C around 0 37.9%

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

      1. neg-mul-1 37.9%

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

   9. Simplified 37.9%

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

   if 1.31999999999999998e154 < B

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in A around -inf 0.0%

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

   5. Taylor expanded in B around inf 6.9%

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

      1. un-div-inv 6.9%

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

   7. Applied egg-rr 6.9%

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

4. Final simplification 28.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -1.5 \cdot 10^{-9}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;B \leq 3.8 \cdot 10^{-65}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - \left(A \cdot C\right) \cdot 4\right)\right) \cdot \left(2 \cdot A\right)\right)}}{B \cdot B - \left(A \cdot C\right) \cdot 4}\\ \mathbf{elif}\;B \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(B \cdot \left(F \cdot \left(\left(A \cdot C\right) \cdot 4 - B \cdot B\right)\right)\right)}}{B \cdot B - \left(A \cdot C\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{F \cdot A}}{B}\\ \end{array} \]

Alternative 9: 24.2% accurate, 4.7× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := \sqrt{F \cdot A}\\ t_1 := \left(A \cdot C\right) \cdot 4\\ t_2 := B \cdot B - t_1\\ \mathbf{if}\;B \leq -5.4 \cdot 10^{+148}:\\ \;\;\;\;2 \cdot \left(t_0 \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -2.2 \cdot 10^{-55}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(B \cdot \left(F \cdot t_2\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 2.5 \cdot 10^{-96}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(2 \cdot A\right) \cdot \left(-4 \cdot \left(A \cdot \left(F \cdot C\right)\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 3.4 \cdot 10^{+153}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(B \cdot \left(F \cdot \left(t_1 - B \cdot B\right)\right)\right)}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{t_0}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (sqrt (* F A))) (t_1 (* (* A C) 4.0)) (t_2 (- (* B B) t_1)))
   (if (<= B -5.4e+148)
     (* 2.0 (* t_0 (/ 1.0 B)))
     (if (<= B -2.2e-55)
       (- (/ (sqrt (* 2.0 (* B (* F t_2)))) t_2))
       (if (<= B 2.5e-96)
         (/ (- (sqrt (* 2.0 (* (* 2.0 A) (* -4.0 (* A (* F C))))))) t_2)
         (if (<= B 3.4e+153)
           (/ (- (sqrt (* 2.0 (* B (* F (- t_1 (* B B))))))) t_2)
           (* -2.0 (/ t_0 B))))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = sqrt((F * A));
	double t_1 = (A * C) * 4.0;
	double t_2 = (B * B) - t_1;
	double tmp;
	if (B <= -5.4e+148) {
		tmp = 2.0 * (t_0 * (1.0 / B));
	} else if (B <= -2.2e-55) {
		tmp = -(sqrt((2.0 * (B * (F * t_2)))) / t_2);
	} else if (B <= 2.5e-96) {
		tmp = -sqrt((2.0 * ((2.0 * A) * (-4.0 * (A * (F * C)))))) / t_2;
	} else if (B <= 3.4e+153) {
		tmp = -sqrt((2.0 * (B * (F * (t_1 - (B * B)))))) / t_2;
	} else {
		tmp = -2.0 * (t_0 / B);
	}
	return tmp;
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this function.
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
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = sqrt((f * a))
    t_1 = (a * c) * 4.0d0
    t_2 = (b * b) - t_1
    if (b <= (-5.4d+148)) then
        tmp = 2.0d0 * (t_0 * (1.0d0 / b))
    else if (b <= (-2.2d-55)) then
        tmp = -(sqrt((2.0d0 * (b * (f * t_2)))) / t_2)
    else if (b <= 2.5d-96) then
        tmp = -sqrt((2.0d0 * ((2.0d0 * a) * ((-4.0d0) * (a * (f * c)))))) / t_2
    else if (b <= 3.4d+153) then
        tmp = -sqrt((2.0d0 * (b * (f * (t_1 - (b * b)))))) / t_2
    else
        tmp = (-2.0d0) * (t_0 / b)
    end if
    code = tmp
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = Math.sqrt((F * A));
	double t_1 = (A * C) * 4.0;
	double t_2 = (B * B) - t_1;
	double tmp;
	if (B <= -5.4e+148) {
		tmp = 2.0 * (t_0 * (1.0 / B));
	} else if (B <= -2.2e-55) {
		tmp = -(Math.sqrt((2.0 * (B * (F * t_2)))) / t_2);
	} else if (B <= 2.5e-96) {
		tmp = -Math.sqrt((2.0 * ((2.0 * A) * (-4.0 * (A * (F * C)))))) / t_2;
	} else if (B <= 3.4e+153) {
		tmp = -Math.sqrt((2.0 * (B * (F * (t_1 - (B * B)))))) / t_2;
	} else {
		tmp = -2.0 * (t_0 / B);
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = math.sqrt((F * A))
	t_1 = (A * C) * 4.0
	t_2 = (B * B) - t_1
	tmp = 0
	if B <= -5.4e+148:
		tmp = 2.0 * (t_0 * (1.0 / B))
	elif B <= -2.2e-55:
		tmp = -(math.sqrt((2.0 * (B * (F * t_2)))) / t_2)
	elif B <= 2.5e-96:
		tmp = -math.sqrt((2.0 * ((2.0 * A) * (-4.0 * (A * (F * C)))))) / t_2
	elif B <= 3.4e+153:
		tmp = -math.sqrt((2.0 * (B * (F * (t_1 - (B * B)))))) / t_2
	else:
		tmp = -2.0 * (t_0 / B)
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = sqrt(Float64(F * A))
	t_1 = Float64(Float64(A * C) * 4.0)
	t_2 = Float64(Float64(B * B) - t_1)
	tmp = 0.0
	if (B <= -5.4e+148)
		tmp = Float64(2.0 * Float64(t_0 * Float64(1.0 / B)));
	elseif (B <= -2.2e-55)
		tmp = Float64(-Float64(sqrt(Float64(2.0 * Float64(B * Float64(F * t_2)))) / t_2));
	elseif (B <= 2.5e-96)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(2.0 * A) * Float64(-4.0 * Float64(A * Float64(F * C))))))) / t_2);
	elseif (B <= 3.4e+153)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(B * Float64(F * Float64(t_1 - Float64(B * B))))))) / t_2);
	else
		tmp = Float64(-2.0 * Float64(t_0 / B));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = sqrt((F * A));
	t_1 = (A * C) * 4.0;
	t_2 = (B * B) - t_1;
	tmp = 0.0;
	if (B <= -5.4e+148)
		tmp = 2.0 * (t_0 * (1.0 / B));
	elseif (B <= -2.2e-55)
		tmp = -(sqrt((2.0 * (B * (F * t_2)))) / t_2);
	elseif (B <= 2.5e-96)
		tmp = -sqrt((2.0 * ((2.0 * A) * (-4.0 * (A * (F * C)))))) / t_2;
	elseif (B <= 3.4e+153)
		tmp = -sqrt((2.0 * (B * (F * (t_1 - (B * B)))))) / t_2;
	else
		tmp = -2.0 * (t_0 / B);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[Sqrt[N[(F * A), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(A * C), $MachinePrecision] * 4.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(B * B), $MachinePrecision] - t$95$1), $MachinePrecision]}, If[LessEqual[B, -5.4e+148], N[(2.0 * N[(t$95$0 * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, -2.2e-55], (-N[(N[Sqrt[N[(2.0 * N[(B * N[(F * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision]), If[LessEqual[B, 2.5e-96], N[((-N[Sqrt[N[(2.0 * N[(N[(2.0 * A), $MachinePrecision] * N[(-4.0 * N[(A * N[(F * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision], If[LessEqual[B, 3.4e+153], N[((-N[Sqrt[N[(2.0 * N[(B * N[(F * N[(t$95$1 - N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision], N[(-2.0 * N[(t$95$0 / B), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if B < -5.40000000000000038e148

   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. Step-by-step derivation

   3. Simplified 0.0%

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

   4. Taylor expanded in A around -inf 0.0%

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

   5. Taylor expanded in B around -inf 2.7%

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

   if -5.40000000000000038e148 < B < -2.2e-55

   1. Initial program 36.4%

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

   3. Simplified 36.4%

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

   4. Taylor expanded in A around 0 33.8%

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

      1. +-commutative 33.8%

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

      2. unpow2 33.8%

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

      3. unpow2 33.8%

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

      4. hypot-def 35.9%

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

   6. Simplified 35.9%

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

   7. Taylor expanded in B around -inf 32.3%

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

   if -2.2e-55 < B < 2.49999999999999997e-96

   1. Initial program 15.1%

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

   3. Simplified 15.1%

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

   4. Taylor expanded in A around -inf 22.3%

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

   5. Taylor expanded in B around 0 19.1%

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

   if 2.49999999999999997e-96 < B < 3.3999999999999997e153

   1. Initial program 39.1%

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

   3. Simplified 39.1%

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

   4. Taylor expanded in A around 0 39.5%

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

      1. +-commutative 39.5%

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

      2. unpow2 39.5%

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

      3. unpow2 39.5%

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

      4. hypot-def 39.8%

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

   6. Simplified 39.8%

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

   7. Taylor expanded in C around 0 36.5%

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

      1. neg-mul-1 36.5%

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

   9. Simplified 36.5%

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

   if 3.3999999999999997e153 < B

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in A around -inf 0.0%

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

   5. Taylor expanded in B around inf 6.9%

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

      1. un-div-inv 6.9%

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

   7. Applied egg-rr 6.9%

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

4. Final simplification 21.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -5.4 \cdot 10^{+148}:\\ \;\;\;\;2 \cdot \left(\sqrt{F \cdot A} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -2.2 \cdot 10^{-55}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(B \cdot \left(F \cdot \left(B \cdot B - \left(A \cdot C\right) \cdot 4\right)\right)\right)}}{B \cdot B - \left(A \cdot C\right) \cdot 4}\\ \mathbf{elif}\;B \leq 2.5 \cdot 10^{-96}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(2 \cdot A\right) \cdot \left(-4 \cdot \left(A \cdot \left(F \cdot C\right)\right)\right)\right)}}{B \cdot B - \left(A \cdot C\right) \cdot 4}\\ \mathbf{elif}\;B \leq 3.4 \cdot 10^{+153}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(B \cdot \left(F \cdot \left(\left(A \cdot C\right) \cdot 4 - B \cdot B\right)\right)\right)}}{B \cdot B - \left(A \cdot C\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{F \cdot A}}{B}\\ \end{array} \]

Alternative 10: 29.6% accurate, 4.7× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right) + B \cdot B\\ t_1 := \sqrt{F \cdot A}\\ t_2 := \left(A \cdot C\right) \cdot 4\\ t_3 := B \cdot B - t_2\\ \mathbf{if}\;B \leq -1.95 \cdot 10^{+146}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -2.8 \cdot 10^{-8}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(B \cdot \left(F \cdot t_3\right)\right)}}{t_3}\\ \mathbf{elif}\;B \leq 5.6 \cdot 10^{-65}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 1.28 \cdot 10^{+154}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(B \cdot \left(F \cdot \left(t_2 - B \cdot B\right)\right)\right)}}{t_3}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{t_1}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (+ (* -4.0 (* A C)) (* B B)))
        (t_1 (sqrt (* F A)))
        (t_2 (* (* A C) 4.0))
        (t_3 (- (* B B) t_2)))
   (if (<= B -1.95e+146)
     (* 2.0 (* t_1 (/ 1.0 B)))
     (if (<= B -2.8e-8)
       (- (/ (sqrt (* 2.0 (* B (* F t_3)))) t_3))
       (if (<= B 5.6e-65)
         (/ (- (sqrt (* 2.0 (* t_0 (* F (* 2.0 A)))))) t_0)
         (if (<= B 1.28e+154)
           (/ (- (sqrt (* 2.0 (* B (* F (- t_2 (* B B))))))) t_3)
           (* -2.0 (/ t_1 B))))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (-4.0 * (A * C)) + (B * B);
	double t_1 = sqrt((F * A));
	double t_2 = (A * C) * 4.0;
	double t_3 = (B * B) - t_2;
	double tmp;
	if (B <= -1.95e+146) {
		tmp = 2.0 * (t_1 * (1.0 / B));
	} else if (B <= -2.8e-8) {
		tmp = -(sqrt((2.0 * (B * (F * t_3)))) / t_3);
	} else if (B <= 5.6e-65) {
		tmp = -sqrt((2.0 * (t_0 * (F * (2.0 * A))))) / t_0;
	} else if (B <= 1.28e+154) {
		tmp = -sqrt((2.0 * (B * (F * (t_2 - (B * B)))))) / t_3;
	} else {
		tmp = -2.0 * (t_1 / B);
	}
	return tmp;
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this function.
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
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = ((-4.0d0) * (a * c)) + (b * b)
    t_1 = sqrt((f * a))
    t_2 = (a * c) * 4.0d0
    t_3 = (b * b) - t_2
    if (b <= (-1.95d+146)) then
        tmp = 2.0d0 * (t_1 * (1.0d0 / b))
    else if (b <= (-2.8d-8)) then
        tmp = -(sqrt((2.0d0 * (b * (f * t_3)))) / t_3)
    else if (b <= 5.6d-65) then
        tmp = -sqrt((2.0d0 * (t_0 * (f * (2.0d0 * a))))) / t_0
    else if (b <= 1.28d+154) then
        tmp = -sqrt((2.0d0 * (b * (f * (t_2 - (b * b)))))) / t_3
    else
        tmp = (-2.0d0) * (t_1 / b)
    end if
    code = tmp
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = (-4.0 * (A * C)) + (B * B);
	double t_1 = Math.sqrt((F * A));
	double t_2 = (A * C) * 4.0;
	double t_3 = (B * B) - t_2;
	double tmp;
	if (B <= -1.95e+146) {
		tmp = 2.0 * (t_1 * (1.0 / B));
	} else if (B <= -2.8e-8) {
		tmp = -(Math.sqrt((2.0 * (B * (F * t_3)))) / t_3);
	} else if (B <= 5.6e-65) {
		tmp = -Math.sqrt((2.0 * (t_0 * (F * (2.0 * A))))) / t_0;
	} else if (B <= 1.28e+154) {
		tmp = -Math.sqrt((2.0 * (B * (F * (t_2 - (B * B)))))) / t_3;
	} else {
		tmp = -2.0 * (t_1 / B);
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (-4.0 * (A * C)) + (B * B)
	t_1 = math.sqrt((F * A))
	t_2 = (A * C) * 4.0
	t_3 = (B * B) - t_2
	tmp = 0
	if B <= -1.95e+146:
		tmp = 2.0 * (t_1 * (1.0 / B))
	elif B <= -2.8e-8:
		tmp = -(math.sqrt((2.0 * (B * (F * t_3)))) / t_3)
	elif B <= 5.6e-65:
		tmp = -math.sqrt((2.0 * (t_0 * (F * (2.0 * A))))) / t_0
	elif B <= 1.28e+154:
		tmp = -math.sqrt((2.0 * (B * (F * (t_2 - (B * B)))))) / t_3
	else:
		tmp = -2.0 * (t_1 / B)
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(-4.0 * Float64(A * C)) + Float64(B * B))
	t_1 = sqrt(Float64(F * A))
	t_2 = Float64(Float64(A * C) * 4.0)
	t_3 = Float64(Float64(B * B) - t_2)
	tmp = 0.0
	if (B <= -1.95e+146)
		tmp = Float64(2.0 * Float64(t_1 * Float64(1.0 / B)));
	elseif (B <= -2.8e-8)
		tmp = Float64(-Float64(sqrt(Float64(2.0 * Float64(B * Float64(F * t_3)))) / t_3));
	elseif (B <= 5.6e-65)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_0 * Float64(F * Float64(2.0 * A)))))) / t_0);
	elseif (B <= 1.28e+154)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(B * Float64(F * Float64(t_2 - Float64(B * B))))))) / t_3);
	else
		tmp = Float64(-2.0 * Float64(t_1 / B));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (-4.0 * (A * C)) + (B * B);
	t_1 = sqrt((F * A));
	t_2 = (A * C) * 4.0;
	t_3 = (B * B) - t_2;
	tmp = 0.0;
	if (B <= -1.95e+146)
		tmp = 2.0 * (t_1 * (1.0 / B));
	elseif (B <= -2.8e-8)
		tmp = -(sqrt((2.0 * (B * (F * t_3)))) / t_3);
	elseif (B <= 5.6e-65)
		tmp = -sqrt((2.0 * (t_0 * (F * (2.0 * A))))) / t_0;
	elseif (B <= 1.28e+154)
		tmp = -sqrt((2.0 * (B * (F * (t_2 - (B * B)))))) / t_3;
	else
		tmp = -2.0 * (t_1 / B);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(F * A), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(A * C), $MachinePrecision] * 4.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(B * B), $MachinePrecision] - t$95$2), $MachinePrecision]}, If[LessEqual[B, -1.95e+146], N[(2.0 * N[(t$95$1 * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, -2.8e-8], (-N[(N[Sqrt[N[(2.0 * N[(B * N[(F * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision]), If[LessEqual[B, 5.6e-65], N[((-N[Sqrt[N[(2.0 * N[(t$95$0 * N[(F * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[B, 1.28e+154], N[((-N[Sqrt[N[(2.0 * N[(B * N[(F * N[(t$95$2 - N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$3), $MachinePrecision], N[(-2.0 * N[(t$95$1 / B), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if B < -1.95e146

   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. Step-by-step derivation

   3. Simplified 0.0%

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

   4. Taylor expanded in A around -inf 0.0%

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

   5. Taylor expanded in B around -inf 2.7%

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

   if -1.95e146 < B < -2.7999999999999999e-8

   1. Initial program 34.0%

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

   3. Simplified 34.0%

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

   4. Taylor expanded in A around 0 30.9%

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

      1. +-commutative 30.9%

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

      2. unpow2 30.9%

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

      3. unpow2 30.9%

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

      4. hypot-def 33.1%

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

   6. Simplified 33.1%

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

   7. Taylor expanded in B around -inf 31.3%

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

   if -2.7999999999999999e-8 < B < 5.6000000000000001e-65

   1. Initial program 20.1%

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

   3. Simplified 20.1%

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

   4. Taylor expanded in A around -inf 21.6%

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

      1. distribute-frac-neg 21.6%

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

      2. associate-*l* 21.5%

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

      3. cancel-sign-sub-inv 21.5%

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

      4. metadata-eval 21.5%

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

      5. cancel-sign-sub-inv 21.5%

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

      6. metadata-eval 21.5%

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

   6. Applied egg-rr 21.5%

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

   if 5.6000000000000001e-65 < B < 1.2800000000000001e154

   1. Initial program 37.3%

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

   3. Simplified 37.3%

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

   4. Taylor expanded in A around 0 37.7%

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

      1. +-commutative 37.7%

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

      2. unpow2 37.7%

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

      3. unpow2 37.7%

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

      4. hypot-def 38.0%

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

   6. Simplified 38.0%

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

   7. Taylor expanded in C around 0 37.9%

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

      1. neg-mul-1 37.9%

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

   9. Simplified 37.9%

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

   if 1.2800000000000001e154 < B

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in A around -inf 0.0%

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

   5. Taylor expanded in B around inf 6.9%

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

      1. un-div-inv 6.9%

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

   7. Applied egg-rr 6.9%

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

4. Final simplification 21.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -1.95 \cdot 10^{+146}:\\ \;\;\;\;2 \cdot \left(\sqrt{F \cdot A} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -2.8 \cdot 10^{-8}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(B \cdot \left(F \cdot \left(B \cdot B - \left(A \cdot C\right) \cdot 4\right)\right)\right)}}{B \cdot B - \left(A \cdot C\right) \cdot 4}\\ \mathbf{elif}\;B \leq 5.6 \cdot 10^{-65}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(-4 \cdot \left(A \cdot C\right) + B \cdot B\right) \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{-4 \cdot \left(A \cdot C\right) + B \cdot B}\\ \mathbf{elif}\;B \leq 1.28 \cdot 10^{+154}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(B \cdot \left(F \cdot \left(\left(A \cdot C\right) \cdot 4 - B \cdot B\right)\right)\right)}}{B \cdot B - \left(A \cdot C\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{F \cdot A}}{B}\\ \end{array} \]

Alternative 11: 29.5% accurate, 4.7× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := \sqrt{F \cdot A}\\ t_1 := \left(A \cdot C\right) \cdot 4\\ t_2 := B \cdot B - t_1\\ t_3 := F \cdot t_2\\ \mathbf{if}\;B \leq -4.1 \cdot 10^{+147}:\\ \;\;\;\;2 \cdot \left(t_0 \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -1.8 \cdot 10^{-14}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(B \cdot t_3\right)}}{t_2}\\ \mathbf{elif}\;B \leq 7.5 \cdot 10^{-65}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_3 \cdot \left(2 \cdot A\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 1.1 \cdot 10^{+150}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(B \cdot \left(F \cdot \left(t_1 - B \cdot B\right)\right)\right)}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{t_0}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (sqrt (* F A)))
        (t_1 (* (* A C) 4.0))
        (t_2 (- (* B B) t_1))
        (t_3 (* F t_2)))
   (if (<= B -4.1e+147)
     (* 2.0 (* t_0 (/ 1.0 B)))
     (if (<= B -1.8e-14)
       (- (/ (sqrt (* 2.0 (* B t_3))) t_2))
       (if (<= B 7.5e-65)
         (/ (- (sqrt (* 2.0 (* t_3 (* 2.0 A))))) t_2)
         (if (<= B 1.1e+150)
           (/ (- (sqrt (* 2.0 (* B (* F (- t_1 (* B B))))))) t_2)
           (* -2.0 (/ t_0 B))))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = sqrt((F * A));
	double t_1 = (A * C) * 4.0;
	double t_2 = (B * B) - t_1;
	double t_3 = F * t_2;
	double tmp;
	if (B <= -4.1e+147) {
		tmp = 2.0 * (t_0 * (1.0 / B));
	} else if (B <= -1.8e-14) {
		tmp = -(sqrt((2.0 * (B * t_3))) / t_2);
	} else if (B <= 7.5e-65) {
		tmp = -sqrt((2.0 * (t_3 * (2.0 * A)))) / t_2;
	} else if (B <= 1.1e+150) {
		tmp = -sqrt((2.0 * (B * (F * (t_1 - (B * B)))))) / t_2;
	} else {
		tmp = -2.0 * (t_0 / B);
	}
	return tmp;
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this function.
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
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = sqrt((f * a))
    t_1 = (a * c) * 4.0d0
    t_2 = (b * b) - t_1
    t_3 = f * t_2
    if (b <= (-4.1d+147)) then
        tmp = 2.0d0 * (t_0 * (1.0d0 / b))
    else if (b <= (-1.8d-14)) then
        tmp = -(sqrt((2.0d0 * (b * t_3))) / t_2)
    else if (b <= 7.5d-65) then
        tmp = -sqrt((2.0d0 * (t_3 * (2.0d0 * a)))) / t_2
    else if (b <= 1.1d+150) then
        tmp = -sqrt((2.0d0 * (b * (f * (t_1 - (b * b)))))) / t_2
    else
        tmp = (-2.0d0) * (t_0 / b)
    end if
    code = tmp
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = Math.sqrt((F * A));
	double t_1 = (A * C) * 4.0;
	double t_2 = (B * B) - t_1;
	double t_3 = F * t_2;
	double tmp;
	if (B <= -4.1e+147) {
		tmp = 2.0 * (t_0 * (1.0 / B));
	} else if (B <= -1.8e-14) {
		tmp = -(Math.sqrt((2.0 * (B * t_3))) / t_2);
	} else if (B <= 7.5e-65) {
		tmp = -Math.sqrt((2.0 * (t_3 * (2.0 * A)))) / t_2;
	} else if (B <= 1.1e+150) {
		tmp = -Math.sqrt((2.0 * (B * (F * (t_1 - (B * B)))))) / t_2;
	} else {
		tmp = -2.0 * (t_0 / B);
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = math.sqrt((F * A))
	t_1 = (A * C) * 4.0
	t_2 = (B * B) - t_1
	t_3 = F * t_2
	tmp = 0
	if B <= -4.1e+147:
		tmp = 2.0 * (t_0 * (1.0 / B))
	elif B <= -1.8e-14:
		tmp = -(math.sqrt((2.0 * (B * t_3))) / t_2)
	elif B <= 7.5e-65:
		tmp = -math.sqrt((2.0 * (t_3 * (2.0 * A)))) / t_2
	elif B <= 1.1e+150:
		tmp = -math.sqrt((2.0 * (B * (F * (t_1 - (B * B)))))) / t_2
	else:
		tmp = -2.0 * (t_0 / B)
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = sqrt(Float64(F * A))
	t_1 = Float64(Float64(A * C) * 4.0)
	t_2 = Float64(Float64(B * B) - t_1)
	t_3 = Float64(F * t_2)
	tmp = 0.0
	if (B <= -4.1e+147)
		tmp = Float64(2.0 * Float64(t_0 * Float64(1.0 / B)));
	elseif (B <= -1.8e-14)
		tmp = Float64(-Float64(sqrt(Float64(2.0 * Float64(B * t_3))) / t_2));
	elseif (B <= 7.5e-65)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_3 * Float64(2.0 * A))))) / t_2);
	elseif (B <= 1.1e+150)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(B * Float64(F * Float64(t_1 - Float64(B * B))))))) / t_2);
	else
		tmp = Float64(-2.0 * Float64(t_0 / B));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = sqrt((F * A));
	t_1 = (A * C) * 4.0;
	t_2 = (B * B) - t_1;
	t_3 = F * t_2;
	tmp = 0.0;
	if (B <= -4.1e+147)
		tmp = 2.0 * (t_0 * (1.0 / B));
	elseif (B <= -1.8e-14)
		tmp = -(sqrt((2.0 * (B * t_3))) / t_2);
	elseif (B <= 7.5e-65)
		tmp = -sqrt((2.0 * (t_3 * (2.0 * A)))) / t_2;
	elseif (B <= 1.1e+150)
		tmp = -sqrt((2.0 * (B * (F * (t_1 - (B * B)))))) / t_2;
	else
		tmp = -2.0 * (t_0 / B);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[Sqrt[N[(F * A), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(A * C), $MachinePrecision] * 4.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(B * B), $MachinePrecision] - t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(F * t$95$2), $MachinePrecision]}, If[LessEqual[B, -4.1e+147], N[(2.0 * N[(t$95$0 * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, -1.8e-14], (-N[(N[Sqrt[N[(2.0 * N[(B * t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision]), If[LessEqual[B, 7.5e-65], N[((-N[Sqrt[N[(2.0 * N[(t$95$3 * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision], If[LessEqual[B, 1.1e+150], N[((-N[Sqrt[N[(2.0 * N[(B * N[(F * N[(t$95$1 - N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision], N[(-2.0 * N[(t$95$0 / B), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if B < -4.09999999999999966e147

   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. Step-by-step derivation

   3. Simplified 0.0%

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

   4. Taylor expanded in A around -inf 0.0%

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

   5. Taylor expanded in B around -inf 2.7%

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

   if -4.09999999999999966e147 < B < -1.7999999999999999e-14

   1. Initial program 34.0%

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

   3. Simplified 34.0%

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

   4. Taylor expanded in A around 0 30.9%

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

      1. +-commutative 30.9%

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

      2. unpow2 30.9%

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

      3. unpow2 30.9%

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

      4. hypot-def 33.1%

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

   6. Simplified 33.1%

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

   7. Taylor expanded in B around -inf 31.3%

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

   if -1.7999999999999999e-14 < B < 7.5000000000000002e-65

   1. Initial program 20.1%

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

   3. Simplified 20.1%

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

   4. Taylor expanded in A around -inf 21.6%

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

   if 7.5000000000000002e-65 < B < 1.1e150

   1. Initial program 37.3%

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

   3. Simplified 37.3%

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

   4. Taylor expanded in A around 0 37.7%

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

      1. +-commutative 37.7%

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

      2. unpow2 37.7%

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

      3. unpow2 37.7%

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

      4. hypot-def 38.0%

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

   6. Simplified 38.0%

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

   7. Taylor expanded in C around 0 37.9%

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

      1. neg-mul-1 37.9%

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

   9. Simplified 37.9%

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

   if 1.1e150 < B

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in A around -inf 0.0%

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

   5. Taylor expanded in B around inf 6.9%

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

      1. un-div-inv 6.9%

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

   7. Applied egg-rr 6.9%

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

4. Final simplification 21.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -4.1 \cdot 10^{+147}:\\ \;\;\;\;2 \cdot \left(\sqrt{F \cdot A} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -1.8 \cdot 10^{-14}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(B \cdot \left(F \cdot \left(B \cdot B - \left(A \cdot C\right) \cdot 4\right)\right)\right)}}{B \cdot B - \left(A \cdot C\right) \cdot 4}\\ \mathbf{elif}\;B \leq 7.5 \cdot 10^{-65}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - \left(A \cdot C\right) \cdot 4\right)\right) \cdot \left(2 \cdot A\right)\right)}}{B \cdot B - \left(A \cdot C\right) \cdot 4}\\ \mathbf{elif}\;B \leq 1.1 \cdot 10^{+150}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(B \cdot \left(F \cdot \left(\left(A \cdot C\right) \cdot 4 - B \cdot B\right)\right)\right)}}{B \cdot B - \left(A \cdot C\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{F \cdot A}}{B}\\ \end{array} \]

Alternative 12: 29.6% accurate, 4.7× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := \sqrt{F \cdot A}\\ t_1 := \left(A \cdot C\right) \cdot 4\\ t_2 := B \cdot B - t_1\\ t_3 := F \cdot t_2\\ \mathbf{if}\;B \leq -5.7 \cdot 10^{+146}:\\ \;\;\;\;2 \cdot \left(t_0 \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -2.7 \cdot 10^{-6}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_3 \cdot \left(A + \left(B + C\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 7.5 \cdot 10^{-65}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_3 \cdot \left(2 \cdot A\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 7.6 \cdot 10^{+153}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(B \cdot \left(F \cdot \left(t_1 - B \cdot B\right)\right)\right)}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{t_0}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (sqrt (* F A)))
        (t_1 (* (* A C) 4.0))
        (t_2 (- (* B B) t_1))
        (t_3 (* F t_2)))
   (if (<= B -5.7e+146)
     (* 2.0 (* t_0 (/ 1.0 B)))
     (if (<= B -2.7e-6)
       (/ (- (sqrt (* 2.0 (* t_3 (+ A (+ B C)))))) t_2)
       (if (<= B 7.5e-65)
         (/ (- (sqrt (* 2.0 (* t_3 (* 2.0 A))))) t_2)
         (if (<= B 7.6e+153)
           (/ (- (sqrt (* 2.0 (* B (* F (- t_1 (* B B))))))) t_2)
           (* -2.0 (/ t_0 B))))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = sqrt((F * A));
	double t_1 = (A * C) * 4.0;
	double t_2 = (B * B) - t_1;
	double t_3 = F * t_2;
	double tmp;
	if (B <= -5.7e+146) {
		tmp = 2.0 * (t_0 * (1.0 / B));
	} else if (B <= -2.7e-6) {
		tmp = -sqrt((2.0 * (t_3 * (A + (B + C))))) / t_2;
	} else if (B <= 7.5e-65) {
		tmp = -sqrt((2.0 * (t_3 * (2.0 * A)))) / t_2;
	} else if (B <= 7.6e+153) {
		tmp = -sqrt((2.0 * (B * (F * (t_1 - (B * B)))))) / t_2;
	} else {
		tmp = -2.0 * (t_0 / B);
	}
	return tmp;
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this function.
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
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = sqrt((f * a))
    t_1 = (a * c) * 4.0d0
    t_2 = (b * b) - t_1
    t_3 = f * t_2
    if (b <= (-5.7d+146)) then
        tmp = 2.0d0 * (t_0 * (1.0d0 / b))
    else if (b <= (-2.7d-6)) then
        tmp = -sqrt((2.0d0 * (t_3 * (a + (b + c))))) / t_2
    else if (b <= 7.5d-65) then
        tmp = -sqrt((2.0d0 * (t_3 * (2.0d0 * a)))) / t_2
    else if (b <= 7.6d+153) then
        tmp = -sqrt((2.0d0 * (b * (f * (t_1 - (b * b)))))) / t_2
    else
        tmp = (-2.0d0) * (t_0 / b)
    end if
    code = tmp
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = Math.sqrt((F * A));
	double t_1 = (A * C) * 4.0;
	double t_2 = (B * B) - t_1;
	double t_3 = F * t_2;
	double tmp;
	if (B <= -5.7e+146) {
		tmp = 2.0 * (t_0 * (1.0 / B));
	} else if (B <= -2.7e-6) {
		tmp = -Math.sqrt((2.0 * (t_3 * (A + (B + C))))) / t_2;
	} else if (B <= 7.5e-65) {
		tmp = -Math.sqrt((2.0 * (t_3 * (2.0 * A)))) / t_2;
	} else if (B <= 7.6e+153) {
		tmp = -Math.sqrt((2.0 * (B * (F * (t_1 - (B * B)))))) / t_2;
	} else {
		tmp = -2.0 * (t_0 / B);
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = math.sqrt((F * A))
	t_1 = (A * C) * 4.0
	t_2 = (B * B) - t_1
	t_3 = F * t_2
	tmp = 0
	if B <= -5.7e+146:
		tmp = 2.0 * (t_0 * (1.0 / B))
	elif B <= -2.7e-6:
		tmp = -math.sqrt((2.0 * (t_3 * (A + (B + C))))) / t_2
	elif B <= 7.5e-65:
		tmp = -math.sqrt((2.0 * (t_3 * (2.0 * A)))) / t_2
	elif B <= 7.6e+153:
		tmp = -math.sqrt((2.0 * (B * (F * (t_1 - (B * B)))))) / t_2
	else:
		tmp = -2.0 * (t_0 / B)
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = sqrt(Float64(F * A))
	t_1 = Float64(Float64(A * C) * 4.0)
	t_2 = Float64(Float64(B * B) - t_1)
	t_3 = Float64(F * t_2)
	tmp = 0.0
	if (B <= -5.7e+146)
		tmp = Float64(2.0 * Float64(t_0 * Float64(1.0 / B)));
	elseif (B <= -2.7e-6)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_3 * Float64(A + Float64(B + C)))))) / t_2);
	elseif (B <= 7.5e-65)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_3 * Float64(2.0 * A))))) / t_2);
	elseif (B <= 7.6e+153)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(B * Float64(F * Float64(t_1 - Float64(B * B))))))) / t_2);
	else
		tmp = Float64(-2.0 * Float64(t_0 / B));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = sqrt((F * A));
	t_1 = (A * C) * 4.0;
	t_2 = (B * B) - t_1;
	t_3 = F * t_2;
	tmp = 0.0;
	if (B <= -5.7e+146)
		tmp = 2.0 * (t_0 * (1.0 / B));
	elseif (B <= -2.7e-6)
		tmp = -sqrt((2.0 * (t_3 * (A + (B + C))))) / t_2;
	elseif (B <= 7.5e-65)
		tmp = -sqrt((2.0 * (t_3 * (2.0 * A)))) / t_2;
	elseif (B <= 7.6e+153)
		tmp = -sqrt((2.0 * (B * (F * (t_1 - (B * B)))))) / t_2;
	else
		tmp = -2.0 * (t_0 / B);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[Sqrt[N[(F * A), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(A * C), $MachinePrecision] * 4.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(B * B), $MachinePrecision] - t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(F * t$95$2), $MachinePrecision]}, If[LessEqual[B, -5.7e+146], N[(2.0 * N[(t$95$0 * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, -2.7e-6], N[((-N[Sqrt[N[(2.0 * N[(t$95$3 * N[(A + N[(B + C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision], If[LessEqual[B, 7.5e-65], N[((-N[Sqrt[N[(2.0 * N[(t$95$3 * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision], If[LessEqual[B, 7.6e+153], N[((-N[Sqrt[N[(2.0 * N[(B * N[(F * N[(t$95$1 - N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision], N[(-2.0 * N[(t$95$0 / B), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if B < -5.6999999999999999e146

   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. Step-by-step derivation

   3. Simplified 0.0%

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

   4. Taylor expanded in A around -inf 0.0%

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

   5. Taylor expanded in B around -inf 2.7%

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

   if -5.6999999999999999e146 < B < -2.69999999999999998e-6

   1. Initial program 35.4%

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

   3. Simplified 35.4%

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

   4. Taylor expanded in B around -inf 32.7%

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

   if -2.69999999999999998e-6 < B < 7.5000000000000002e-65

   1. Initial program 19.8%

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

   3. Simplified 19.8%

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

   4. Taylor expanded in A around -inf 21.2%

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

   if 7.5000000000000002e-65 < B < 7.59999999999999933e153

   1. Initial program 37.3%

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

   3. Simplified 37.3%

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

   4. Taylor expanded in A around 0 37.7%

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

      1. +-commutative 37.7%

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

      2. unpow2 37.7%

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

      3. unpow2 37.7%

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

      4. hypot-def 38.0%

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

   6. Simplified 38.0%

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

   7. Taylor expanded in C around 0 37.9%

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

      1. neg-mul-1 37.9%

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

   9. Simplified 37.9%

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

   if 7.59999999999999933e153 < B

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in A around -inf 0.0%

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

   5. Taylor expanded in B around inf 6.9%

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

      1. un-div-inv 6.9%

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

   7. Applied egg-rr 6.9%

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

4. Final simplification 21.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -5.7 \cdot 10^{+146}:\\ \;\;\;\;2 \cdot \left(\sqrt{F \cdot A} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -2.7 \cdot 10^{-6}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - \left(A \cdot C\right) \cdot 4\right)\right) \cdot \left(A + \left(B + C\right)\right)\right)}}{B \cdot B - \left(A \cdot C\right) \cdot 4}\\ \mathbf{elif}\;B \leq 7.5 \cdot 10^{-65}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - \left(A \cdot C\right) \cdot 4\right)\right) \cdot \left(2 \cdot A\right)\right)}}{B \cdot B - \left(A \cdot C\right) \cdot 4}\\ \mathbf{elif}\;B \leq 7.6 \cdot 10^{+153}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(B \cdot \left(F \cdot \left(\left(A \cdot C\right) \cdot 4 - B \cdot B\right)\right)\right)}}{B \cdot B - \left(A \cdot C\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{F \cdot A}}{B}\\ \end{array} \]

Alternative 13: 23.7% accurate, 4.9× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - \left(A \cdot C\right) \cdot 4\\ \mathbf{if}\;B \leq -5.8 \cdot 10^{+149}:\\ \;\;\;\;2 \cdot \left(\sqrt{F \cdot A} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -1.55 \cdot 10^{-55}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(B \cdot \left(F \cdot t_0\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 6 \cdot 10^{-6}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(2 \cdot A\right) \cdot \left(-4 \cdot \left(A \cdot \left(F \cdot C\right)\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{{\left(F \cdot A\right)}^{0.5}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* (* A C) 4.0))))
   (if (<= B -5.8e+149)
     (* 2.0 (* (sqrt (* F A)) (/ 1.0 B)))
     (if (<= B -1.55e-55)
       (- (/ (sqrt (* 2.0 (* B (* F t_0)))) t_0))
       (if (<= B 6e-6)
         (/ (- (sqrt (* 2.0 (* (* 2.0 A) (* -4.0 (* A (* F C))))))) t_0)
         (* -2.0 (/ (pow (* F A) 0.5) B)))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - ((A * C) * 4.0);
	double tmp;
	if (B <= -5.8e+149) {
		tmp = 2.0 * (sqrt((F * A)) * (1.0 / B));
	} else if (B <= -1.55e-55) {
		tmp = -(sqrt((2.0 * (B * (F * t_0)))) / t_0);
	} else if (B <= 6e-6) {
		tmp = -sqrt((2.0 * ((2.0 * A) * (-4.0 * (A * (F * C)))))) / t_0;
	} else {
		tmp = -2.0 * (pow((F * A), 0.5) / B);
	}
	return tmp;
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this function.
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
    real(8) :: tmp
    t_0 = (b * b) - ((a * c) * 4.0d0)
    if (b <= (-5.8d+149)) then
        tmp = 2.0d0 * (sqrt((f * a)) * (1.0d0 / b))
    else if (b <= (-1.55d-55)) then
        tmp = -(sqrt((2.0d0 * (b * (f * t_0)))) / t_0)
    else if (b <= 6d-6) then
        tmp = -sqrt((2.0d0 * ((2.0d0 * a) * ((-4.0d0) * (a * (f * c)))))) / t_0
    else
        tmp = (-2.0d0) * (((f * a) ** 0.5d0) / b)
    end if
    code = tmp
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - ((A * C) * 4.0);
	double tmp;
	if (B <= -5.8e+149) {
		tmp = 2.0 * (Math.sqrt((F * A)) * (1.0 / B));
	} else if (B <= -1.55e-55) {
		tmp = -(Math.sqrt((2.0 * (B * (F * t_0)))) / t_0);
	} else if (B <= 6e-6) {
		tmp = -Math.sqrt((2.0 * ((2.0 * A) * (-4.0 * (A * (F * C)))))) / t_0;
	} else {
		tmp = -2.0 * (Math.pow((F * A), 0.5) / B);
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (B * B) - ((A * C) * 4.0)
	tmp = 0
	if B <= -5.8e+149:
		tmp = 2.0 * (math.sqrt((F * A)) * (1.0 / B))
	elif B <= -1.55e-55:
		tmp = -(math.sqrt((2.0 * (B * (F * t_0)))) / t_0)
	elif B <= 6e-6:
		tmp = -math.sqrt((2.0 * ((2.0 * A) * (-4.0 * (A * (F * C)))))) / t_0
	else:
		tmp = -2.0 * (math.pow((F * A), 0.5) / B)
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(Float64(A * C) * 4.0))
	tmp = 0.0
	if (B <= -5.8e+149)
		tmp = Float64(2.0 * Float64(sqrt(Float64(F * A)) * Float64(1.0 / B)));
	elseif (B <= -1.55e-55)
		tmp = Float64(-Float64(sqrt(Float64(2.0 * Float64(B * Float64(F * t_0)))) / t_0));
	elseif (B <= 6e-6)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(2.0 * A) * Float64(-4.0 * Float64(A * Float64(F * C))))))) / t_0);
	else
		tmp = Float64(-2.0 * Float64((Float64(F * A) ^ 0.5) / B));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - ((A * C) * 4.0);
	tmp = 0.0;
	if (B <= -5.8e+149)
		tmp = 2.0 * (sqrt((F * A)) * (1.0 / B));
	elseif (B <= -1.55e-55)
		tmp = -(sqrt((2.0 * (B * (F * t_0)))) / t_0);
	elseif (B <= 6e-6)
		tmp = -sqrt((2.0 * ((2.0 * A) * (-4.0 * (A * (F * C)))))) / t_0;
	else
		tmp = -2.0 * (((F * A) ^ 0.5) / B);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(N[(A * C), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -5.8e+149], N[(2.0 * N[(N[Sqrt[N[(F * A), $MachinePrecision]], $MachinePrecision] * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, -1.55e-55], (-N[(N[Sqrt[N[(2.0 * N[(B * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), If[LessEqual[B, 6e-6], N[((-N[Sqrt[N[(2.0 * N[(N[(2.0 * A), $MachinePrecision] * N[(-4.0 * N[(A * N[(F * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(-2.0 * N[(N[Power[N[(F * A), $MachinePrecision], 0.5], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if B < -5.8000000000000004e149

   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. Step-by-step derivation

   3. Simplified 0.0%

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

   4. Taylor expanded in A around -inf 0.0%

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

   5. Taylor expanded in B around -inf 2.7%

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

   if -5.8000000000000004e149 < B < -1.54999999999999998e-55

   1. Initial program 36.4%

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

   3. Simplified 36.4%

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

   4. Taylor expanded in A around 0 33.8%

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

      1. +-commutative 33.8%

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

      2. unpow2 33.8%

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

      3. unpow2 33.8%

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

      4. hypot-def 35.9%

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

   6. Simplified 35.9%

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

   7. Taylor expanded in B around -inf 32.3%

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

   if -1.54999999999999998e-55 < B < 6.0000000000000002e-6

   1. Initial program 20.7%

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

   3. Simplified 20.7%

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

   4. Taylor expanded in A around -inf 22.0%

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

   5. Taylor expanded in B around 0 19.6%

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

   if 6.0000000000000002e-6 < B

   1. Initial program 13.6%

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

   3. Simplified 13.6%

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

   4. Taylor expanded in A around -inf 0.9%

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

   5. Taylor expanded in B around inf 5.0%

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

      1. un-div-inv 5.0%

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

   7. Applied egg-rr 5.0%

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

      1. pow1/2 5.1%

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

   9. Applied egg-rr 5.1%

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

4. Final simplification 16.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -5.8 \cdot 10^{+149}:\\ \;\;\;\;2 \cdot \left(\sqrt{F \cdot A} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -1.55 \cdot 10^{-55}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(B \cdot \left(F \cdot \left(B \cdot B - \left(A \cdot C\right) \cdot 4\right)\right)\right)}}{B \cdot B - \left(A \cdot C\right) \cdot 4}\\ \mathbf{elif}\;B \leq 6 \cdot 10^{-6}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(2 \cdot A\right) \cdot \left(-4 \cdot \left(A \cdot \left(F \cdot C\right)\right)\right)\right)}}{B \cdot B - \left(A \cdot C\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{{\left(F \cdot A\right)}^{0.5}}{B}\\ \end{array} \]

Alternative 14: 22.5% accurate, 4.9× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} \mathbf{if}\;B \leq -3.3 \cdot 10^{+74}:\\ \;\;\;\;2 \cdot \left(\sqrt{F \cdot A} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq 5.7 \cdot 10^{-6}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(2 \cdot A\right) \cdot \left(-4 \cdot \left(A \cdot \left(F \cdot C\right)\right)\right)\right)}}{B \cdot B - \left(A \cdot C\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{{\left(F \cdot A\right)}^{0.5}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (if (<= B -3.3e+74)
   (* 2.0 (* (sqrt (* F A)) (/ 1.0 B)))
   (if (<= B 5.7e-6)
     (/
      (- (sqrt (* 2.0 (* (* 2.0 A) (* -4.0 (* A (* F C)))))))
      (- (* B B) (* (* A C) 4.0)))
     (* -2.0 (/ (pow (* F A) 0.5) B)))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double tmp;
	if (B <= -3.3e+74) {
		tmp = 2.0 * (sqrt((F * A)) * (1.0 / B));
	} else if (B <= 5.7e-6) {
		tmp = -sqrt((2.0 * ((2.0 * A) * (-4.0 * (A * (F * C)))))) / ((B * B) - ((A * C) * 4.0));
	} else {
		tmp = -2.0 * (pow((F * A), 0.5) / B);
	}
	return tmp;
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this function.
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) :: tmp
    if (b <= (-3.3d+74)) then
        tmp = 2.0d0 * (sqrt((f * a)) * (1.0d0 / b))
    else if (b <= 5.7d-6) then
        tmp = -sqrt((2.0d0 * ((2.0d0 * a) * ((-4.0d0) * (a * (f * c)))))) / ((b * b) - ((a * c) * 4.0d0))
    else
        tmp = (-2.0d0) * (((f * a) ** 0.5d0) / b)
    end if
    code = tmp
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double tmp;
	if (B <= -3.3e+74) {
		tmp = 2.0 * (Math.sqrt((F * A)) * (1.0 / B));
	} else if (B <= 5.7e-6) {
		tmp = -Math.sqrt((2.0 * ((2.0 * A) * (-4.0 * (A * (F * C)))))) / ((B * B) - ((A * C) * 4.0));
	} else {
		tmp = -2.0 * (Math.pow((F * A), 0.5) / B);
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	tmp = 0
	if B <= -3.3e+74:
		tmp = 2.0 * (math.sqrt((F * A)) * (1.0 / B))
	elif B <= 5.7e-6:
		tmp = -math.sqrt((2.0 * ((2.0 * A) * (-4.0 * (A * (F * C)))))) / ((B * B) - ((A * C) * 4.0))
	else:
		tmp = -2.0 * (math.pow((F * A), 0.5) / B)
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	tmp = 0.0
	if (B <= -3.3e+74)
		tmp = Float64(2.0 * Float64(sqrt(Float64(F * A)) * Float64(1.0 / B)));
	elseif (B <= 5.7e-6)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(2.0 * A) * Float64(-4.0 * Float64(A * Float64(F * C))))))) / Float64(Float64(B * B) - Float64(Float64(A * C) * 4.0)));
	else
		tmp = Float64(-2.0 * Float64((Float64(F * A) ^ 0.5) / B));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (B <= -3.3e+74)
		tmp = 2.0 * (sqrt((F * A)) * (1.0 / B));
	elseif (B <= 5.7e-6)
		tmp = -sqrt((2.0 * ((2.0 * A) * (-4.0 * (A * (F * C)))))) / ((B * B) - ((A * C) * 4.0));
	else
		tmp = -2.0 * (((F * A) ^ 0.5) / B);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := If[LessEqual[B, -3.3e+74], N[(2.0 * N[(N[Sqrt[N[(F * A), $MachinePrecision]], $MachinePrecision] * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 5.7e-6], N[((-N[Sqrt[N[(2.0 * N[(N[(2.0 * A), $MachinePrecision] * N[(-4.0 * N[(A * N[(F * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[(B * B), $MachinePrecision] - N[(N[(A * C), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[Power[N[(F * A), $MachinePrecision], 0.5], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < -3.3000000000000002e74

   1. Initial program 13.4%

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

   3. Simplified 13.4%

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

   4. Taylor expanded in A around -inf 1.6%

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

   5. Taylor expanded in B around -inf 3.1%

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

   if -3.3000000000000002e74 < B < 5.6999999999999996e-6

   1. Initial program 24.7%

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

   3. Simplified 24.7%

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

   4. Taylor expanded in A around -inf 20.5%

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

   5. Taylor expanded in B around 0 18.6%

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

   if 5.6999999999999996e-6 < B

   1. Initial program 13.6%

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

   3. Simplified 13.6%

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

   4. Taylor expanded in A around -inf 0.9%

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

   5. Taylor expanded in B around inf 5.0%

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

      1. un-div-inv 5.0%

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

   7. Applied egg-rr 5.0%

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

      1. pow1/2 5.1%

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

   9. Applied egg-rr 5.1%

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

4. Final simplification 12.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -3.3 \cdot 10^{+74}:\\ \;\;\;\;2 \cdot \left(\sqrt{F \cdot A} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq 5.7 \cdot 10^{-6}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(2 \cdot A\right) \cdot \left(-4 \cdot \left(A \cdot \left(F \cdot C\right)\right)\right)\right)}}{B \cdot B - \left(A \cdot C\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{{\left(F \cdot A\right)}^{0.5}}{B}\\ \end{array} \]

Alternative 15: 17.4% accurate, 5.1× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -8 \cdot 10^{+145}:\\ \;\;\;\;2 \cdot \left(\sqrt{F \cdot A} \cdot \frac{1}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-8 \cdot \left(\left(A \cdot A\right) \cdot \left(F \cdot C\right)\right)\right)}}{B \cdot B - \left(A \cdot C\right) \cdot 4}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (if (<= A -8e+145)
   (* 2.0 (* (sqrt (* F A)) (/ 1.0 B)))
   (/
    (- (sqrt (* 2.0 (* -8.0 (* (* A A) (* F C))))))
    (- (* B B) (* (* A C) 4.0)))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double tmp;
	if (A <= -8e+145) {
		tmp = 2.0 * (sqrt((F * A)) * (1.0 / B));
	} else {
		tmp = -sqrt((2.0 * (-8.0 * ((A * A) * (F * C))))) / ((B * B) - ((A * C) * 4.0));
	}
	return tmp;
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this function.
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) :: tmp
    if (a <= (-8d+145)) then
        tmp = 2.0d0 * (sqrt((f * a)) * (1.0d0 / b))
    else
        tmp = -sqrt((2.0d0 * ((-8.0d0) * ((a * a) * (f * c))))) / ((b * b) - ((a * c) * 4.0d0))
    end if
    code = tmp
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double tmp;
	if (A <= -8e+145) {
		tmp = 2.0 * (Math.sqrt((F * A)) * (1.0 / B));
	} else {
		tmp = -Math.sqrt((2.0 * (-8.0 * ((A * A) * (F * C))))) / ((B * B) - ((A * C) * 4.0));
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	tmp = 0
	if A <= -8e+145:
		tmp = 2.0 * (math.sqrt((F * A)) * (1.0 / B))
	else:
		tmp = -math.sqrt((2.0 * (-8.0 * ((A * A) * (F * C))))) / ((B * B) - ((A * C) * 4.0))
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	tmp = 0.0
	if (A <= -8e+145)
		tmp = Float64(2.0 * Float64(sqrt(Float64(F * A)) * Float64(1.0 / B)));
	else
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(-8.0 * Float64(Float64(A * A) * Float64(F * C)))))) / Float64(Float64(B * B) - Float64(Float64(A * C) * 4.0)));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (A <= -8e+145)
		tmp = 2.0 * (sqrt((F * A)) * (1.0 / B));
	else
		tmp = -sqrt((2.0 * (-8.0 * ((A * A) * (F * C))))) / ((B * B) - ((A * C) * 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := If[LessEqual[A, -8e+145], N[(2.0 * N[(N[Sqrt[N[(F * A), $MachinePrecision]], $MachinePrecision] * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[N[(2.0 * N[(-8.0 * N[(N[(A * A), $MachinePrecision] * N[(F * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[(B * B), $MachinePrecision] - N[(N[(A * C), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -7.9999999999999999e145

   1. Initial program 0.9%

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

   3. Simplified 0.9%

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

   4. Taylor expanded in A around -inf 19.0%

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

   5. Taylor expanded in B around -inf 1.8%

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

   if -7.9999999999999999e145 < A

   1. Initial program 22.2%

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

   3. Simplified 22.2%

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

   4. Taylor expanded in C around 0 21.0%

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

      1. +-commutative 21.0%

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

      2. unpow2 21.0%

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

      3. unpow2 21.0%

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

      4. hypot-def 21.5%

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

   6. Simplified 21.5%

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

   7. Taylor expanded in A around -inf 9.7%

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

      1. unpow2 9.7%

         \[\leadsto \frac{-\sqrt{2 \cdot \left(-8 \cdot \left(\color{blue}{\left(A \cdot A\right)} \cdot \left(C \cdot F\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   9. Simplified 9.7%

      \[\leadsto \frac{-\sqrt{2 \cdot \color{blue}{\left(-8 \cdot \left(\left(A \cdot A\right) \cdot \left(C \cdot F\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 8.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -8 \cdot 10^{+145}:\\ \;\;\;\;2 \cdot \left(\sqrt{F \cdot A} \cdot \frac{1}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-8 \cdot \left(\left(A \cdot A\right) \cdot \left(F \cdot C\right)\right)\right)}}{B \cdot B - \left(A \cdot C\right) \cdot 4}\\ \end{array} \]

Alternative 16: 8.9% accurate, 5.7× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} \mathbf{if}\;B \leq -2 \cdot 10^{-310}:\\ \;\;\;\;2 \cdot \left(\sqrt{F \cdot A} \cdot \frac{1}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{{\left(F \cdot A\right)}^{0.5}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (if (<= B -2e-310)
   (* 2.0 (* (sqrt (* F A)) (/ 1.0 B)))
   (* -2.0 (/ (pow (* F A) 0.5) B))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double tmp;
	if (B <= -2e-310) {
		tmp = 2.0 * (sqrt((F * A)) * (1.0 / B));
	} else {
		tmp = -2.0 * (pow((F * A), 0.5) / B);
	}
	return tmp;
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this function.
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) :: tmp
    if (b <= (-2d-310)) then
        tmp = 2.0d0 * (sqrt((f * a)) * (1.0d0 / b))
    else
        tmp = (-2.0d0) * (((f * a) ** 0.5d0) / b)
    end if
    code = tmp
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double tmp;
	if (B <= -2e-310) {
		tmp = 2.0 * (Math.sqrt((F * A)) * (1.0 / B));
	} else {
		tmp = -2.0 * (Math.pow((F * A), 0.5) / B);
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	tmp = 0
	if B <= -2e-310:
		tmp = 2.0 * (math.sqrt((F * A)) * (1.0 / B))
	else:
		tmp = -2.0 * (math.pow((F * A), 0.5) / B)
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	tmp = 0.0
	if (B <= -2e-310)
		tmp = Float64(2.0 * Float64(sqrt(Float64(F * A)) * Float64(1.0 / B)));
	else
		tmp = Float64(-2.0 * Float64((Float64(F * A) ^ 0.5) / B));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (B <= -2e-310)
		tmp = 2.0 * (sqrt((F * A)) * (1.0 / B));
	else
		tmp = -2.0 * (((F * A) ^ 0.5) / B);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := If[LessEqual[B, -2e-310], N[(2.0 * N[(N[Sqrt[N[(F * A), $MachinePrecision]], $MachinePrecision] * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[Power[N[(F * A), $MachinePrecision], 0.5], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < -1.999999999999994e-310

   1. Initial program 20.8%

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

   3. Simplified 20.8%

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

   4. Taylor expanded in A around -inf 13.3%

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

   5. Taylor expanded in B around -inf 2.8%

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

   if -1.999999999999994e-310 < B

   1. Initial program 18.1%

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

   3. Simplified 18.1%

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

   4. Taylor expanded in A around -inf 9.8%

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

   5. Taylor expanded in B around inf 3.2%

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

      1. un-div-inv 3.2%

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

   7. Applied egg-rr 3.2%

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

      1. pow1/2 3.3%

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

   9. Applied egg-rr 3.3%

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

4. Final simplification 3.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -2 \cdot 10^{-310}:\\ \;\;\;\;2 \cdot \left(\sqrt{F \cdot A} \cdot \frac{1}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{{\left(F \cdot A\right)}^{0.5}}{B}\\ \end{array} \]

Alternative 17: 5.3% accurate, 5.9× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ -2 \cdot \frac{{\left(F \cdot A\right)}^{0.5}}{B} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F) :precision binary64 (* -2.0 (/ (pow (* F A) 0.5) B)))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	return -2.0 * (pow((F * A), 0.5) / B);
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this function.
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
    code = (-2.0d0) * (((f * a) ** 0.5d0) / b)
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	return -2.0 * (Math.pow((F * A), 0.5) / B);
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	return -2.0 * (math.pow((F * A), 0.5) / B)

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	return Float64(-2.0 * Float64((Float64(F * A) ^ 0.5) / B))
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp = code(A, B, C, F)
	tmp = -2.0 * (((F * A) ^ 0.5) / B);
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := N[(-2.0 * N[(N[Power[N[(F * A), $MachinePrecision], 0.5], $MachinePrecision] / B), $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. Step-by-step derivation

3. Simplified 19.6%

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

4. Taylor expanded in A around -inf 11.7%

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

5. Taylor expanded in B around inf 2.0%

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

   1. un-div-inv 2.0%

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

7. Applied egg-rr 2.0%

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

   1. pow1/2 2.1%

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

9. Applied egg-rr 2.1%

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

10. Final simplification 2.1%

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

Alternative 18: 5.2% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: A and C should be sorted in increasing order before calling this function.
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
    code = (-2.0d0) * (sqrt((f * a)) / b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := N[(-2.0 * N[(N[Sqrt[N[(F * A), $MachinePrecision]], $MachinePrecision] / B), $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. Step-by-step derivation

3. Simplified 19.6%

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

4. Taylor expanded in A around -inf 11.7%

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

5. Taylor expanded in B around inf 2.0%

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

   1. un-div-inv 2.0%

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

7. Applied egg-rr 2.0%

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

8. Final simplification 2.0%

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

Reproduce

herbie shell --seed 2023194 
(FPCore (A B C F)
  :name "ABCF->ab-angle b"
  :precision binary64
  (/ (- (sqrt (* (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F)) (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))) (- (pow B 2.0) (* (* 4.0 A) C))))