ABCF->ab-angle b

Percentage Accurate: 19.1% → 45.1%

Time: 25.0s

Alternatives: 14

Speedup: 5.1×

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: 19.1% 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: 45.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} B = |B|\\ [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ t_1 := \frac{-\sqrt{2 \cdot \left(\left(t_0 \cdot F\right) \cdot \left(2 \cdot A\right)\right)}}{t_0}\\ \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-223}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{-129}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \left(-\sqrt{2 \cdot \left(F \cdot \left(A + \left(C - \mathsf{hypot}\left(A - C, B\right)\right)\right)\right)}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{-97}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F \cdot \left(2 \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)\right)}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
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) (* 4.0 (* A C))))
        (t_1 (/ (- (sqrt (* 2.0 (* (* t_0 F) (* 2.0 A))))) t_0)))
   (if (<= (pow B 2.0) 2e-223)
     t_1
     (if (<= (pow B 2.0) 2e-129)
       (/
        (*
         (sqrt (fma B B (* C (* A -4.0))))
         (- (sqrt (* 2.0 (* F (+ A (- C (hypot (- A C) B))))))))
        (fma B B (* A (* C -4.0))))
       (if (<= (pow B 2.0) 5e-97)
         t_1
         (/ (- (sqrt (* F (* 2.0 (- A (hypot A B)))))) B))))))

C

B = abs(B);
assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double t_1 = -sqrt((2.0 * ((t_0 * F) * (2.0 * A)))) / t_0;
	double tmp;
	if (pow(B, 2.0) <= 2e-223) {
		tmp = t_1;
	} else if (pow(B, 2.0) <= 2e-129) {
		tmp = (sqrt(fma(B, B, (C * (A * -4.0)))) * -sqrt((2.0 * (F * (A + (C - hypot((A - C), B))))))) / fma(B, B, (A * (C * -4.0)));
	} else if (pow(B, 2.0) <= 5e-97) {
		tmp = t_1;
	} else {
		tmp = -sqrt((F * (2.0 * (A - hypot(A, B))))) / B;
	}
	return tmp;
}

Julia

B = abs(B)
A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	t_1 = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(t_0 * F) * Float64(2.0 * A))))) / t_0)
	tmp = 0.0
	if ((B ^ 2.0) <= 2e-223)
		tmp = t_1;
	elseif ((B ^ 2.0) <= 2e-129)
		tmp = Float64(Float64(sqrt(fma(B, B, Float64(C * Float64(A * -4.0)))) * Float64(-sqrt(Float64(2.0 * Float64(F * Float64(A + Float64(C - hypot(Float64(A - C), B)))))))) / fma(B, B, Float64(A * Float64(C * -4.0))));
	elseif ((B ^ 2.0) <= 5e-97)
		tmp = t_1;
	else
		tmp = Float64(Float64(-sqrt(Float64(F * Float64(2.0 * Float64(A - hypot(A, B)))))) / B);
	end
	return tmp
end

Wolfram

NOTE: B should be positive before calling this function
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[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-N[Sqrt[N[(2.0 * N[(N[(t$95$0 * F), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision]}, If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 2e-223], t$95$1, If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 2e-129], N[(N[(N[Sqrt[N[(B * B + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * N[(F * N[(A + N[(C - N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / N[(B * B + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 5e-97], t$95$1, N[((-N[Sqrt[N[(F * N[(2.0 * N[(A - N[Sqrt[A ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / B), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B 2) < 1.9999999999999999e-223 or 1.9999999999999999e-129 < (pow.f64 B 2) < 4.9999999999999995e-97

   1. Initial program 17.8%

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

   3. Simplified 17.8%

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

   4. Taylor expanded in A around -inf 27.3%

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

      1. *-commutative 27.3%

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

   6. Simplified 27.3%

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

   if 1.9999999999999999e-223 < (pow.f64 B 2) < 1.9999999999999999e-129

   1. Initial program 26.7%

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

   3. Simplified 32.9%

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

      1. sqrt-prod 50.6%

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

      2. associate-*r* 50.6%

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

      3. *-commutative 50.6%

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

      4. associate-*l* 50.6%

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

      5. associate--r- 50.6%

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

      6. +-commutative 50.6%

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

   5. Applied egg-rr 50.6%

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

      1. hypot-def 44.4%

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

      2. unpow2 44.4%

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

      3. unpow2 44.4%

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

      4. +-commutative 44.4%

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

      5. unpow2 44.4%

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

      6. unpow2 44.4%

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

      7. hypot-def 50.6%

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

   7. Simplified 50.6%

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

   if 4.9999999999999995e-97 < (pow.f64 B 2)

   1. Initial program 13.2%

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

   3. Simplified 13.2%

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

   4. Taylor expanded in C around 0 12.8%

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

      1. mul-1-neg 12.8%

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

      2. *-commutative 12.8%

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

      3. +-commutative 12.8%

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

      4. unpow2 12.8%

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

      5. unpow2 12.8%

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

      6. hypot-def 21.5%

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

   6. Simplified 21.5%

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

      1. associate-*l/ 21.5%

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

   8. Applied egg-rr 21.5%

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

      1. expm1-log1p-u 20.8%

         \[\leadsto -\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}\right)\right)}}{B} \]

      2. expm1-udef 14.3%

         \[\leadsto -\frac{\color{blue}{e^{\mathsf{log1p}\left(\sqrt{2} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}\right)} - 1}}{B} \]

      3. sqrt-unprod 14.3%

         \[\leadsto -\frac{e^{\mathsf{log1p}\left(\color{blue}{\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)\right)}}\right)} - 1}{B} \]

   10. Applied egg-rr 14.3%

       \[\leadsto -\frac{\color{blue}{e^{\mathsf{log1p}\left(\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)\right)}\right)} - 1}}{B} \]
   11. Step-by-step derivation

       1. expm1-def 20.8%

          \[\leadsto -\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)\right)}\right)\right)}}{B} \]

       2. expm1-log1p 21.5%

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

       3. *-commutative 21.5%

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

       4. associate-*l* 21.5%

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

   12. Simplified 21.5%

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

4. Final simplification 25.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-223}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(2 \cdot A\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{-129}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \left(-\sqrt{2 \cdot \left(F \cdot \left(A + \left(C - \mathsf{hypot}\left(A - C, B\right)\right)\right)\right)}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{-97}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(2 \cdot A\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F \cdot \left(2 \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)\right)}}{B}\\ \end{array} \]

Alternative 2: 44.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} B = |B|\\ [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ t_1 := \frac{-\sqrt{2 \cdot \left(\left(t_0 \cdot F\right) \cdot \left(2 \cdot A\right)\right)}}{t_0}\\ t_2 := A - \mathsf{hypot}\left(A, B\right)\\ \mathbf{if}\;B \leq 1.8 \cdot 10^{-104}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 8.5 \cdot 10^{-65}:\\ \;\;\;\;\frac{\sqrt{F \cdot t_2} \cdot \left(B \cdot \left(-\sqrt{2}\right)\right)}{t_0}\\ \mathbf{elif}\;B \leq 7.2 \cdot 10^{-45}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F \cdot \left(2 \cdot t_2\right)}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
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) (* 4.0 (* A C))))
        (t_1 (/ (- (sqrt (* 2.0 (* (* t_0 F) (* 2.0 A))))) t_0))
        (t_2 (- A (hypot A B))))
   (if (<= B 1.8e-104)
     t_1
     (if (<= B 8.5e-65)
       (/ (* (sqrt (* F t_2)) (* B (- (sqrt 2.0)))) t_0)
       (if (<= B 7.2e-45) t_1 (/ (- (sqrt (* F (* 2.0 t_2)))) B))))))

C

B = abs(B);
assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double t_1 = -sqrt((2.0 * ((t_0 * F) * (2.0 * A)))) / t_0;
	double t_2 = A - hypot(A, B);
	double tmp;
	if (B <= 1.8e-104) {
		tmp = t_1;
	} else if (B <= 8.5e-65) {
		tmp = (sqrt((F * t_2)) * (B * -sqrt(2.0))) / t_0;
	} else if (B <= 7.2e-45) {
		tmp = t_1;
	} else {
		tmp = -sqrt((F * (2.0 * t_2))) / B;
	}
	return tmp;
}

Java

B = Math.abs(B);
assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double t_1 = -Math.sqrt((2.0 * ((t_0 * F) * (2.0 * A)))) / t_0;
	double t_2 = A - Math.hypot(A, B);
	double tmp;
	if (B <= 1.8e-104) {
		tmp = t_1;
	} else if (B <= 8.5e-65) {
		tmp = (Math.sqrt((F * t_2)) * (B * -Math.sqrt(2.0))) / t_0;
	} else if (B <= 7.2e-45) {
		tmp = t_1;
	} else {
		tmp = -Math.sqrt((F * (2.0 * t_2))) / B;
	}
	return tmp;
}

Python

B = abs(B)
[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (A * C))
	t_1 = -math.sqrt((2.0 * ((t_0 * F) * (2.0 * A)))) / t_0
	t_2 = A - math.hypot(A, B)
	tmp = 0
	if B <= 1.8e-104:
		tmp = t_1
	elif B <= 8.5e-65:
		tmp = (math.sqrt((F * t_2)) * (B * -math.sqrt(2.0))) / t_0
	elif B <= 7.2e-45:
		tmp = t_1
	else:
		tmp = -math.sqrt((F * (2.0 * t_2))) / B
	return tmp

Julia

B = abs(B)
A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	t_1 = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(t_0 * F) * Float64(2.0 * A))))) / t_0)
	t_2 = Float64(A - hypot(A, B))
	tmp = 0.0
	if (B <= 1.8e-104)
		tmp = t_1;
	elseif (B <= 8.5e-65)
		tmp = Float64(Float64(sqrt(Float64(F * t_2)) * Float64(B * Float64(-sqrt(2.0)))) / t_0);
	elseif (B <= 7.2e-45)
		tmp = t_1;
	else
		tmp = Float64(Float64(-sqrt(Float64(F * Float64(2.0 * t_2)))) / B);
	end
	return tmp
end

MATLAB

B = abs(B)
A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (A * C));
	t_1 = -sqrt((2.0 * ((t_0 * F) * (2.0 * A)))) / t_0;
	t_2 = A - hypot(A, B);
	tmp = 0.0;
	if (B <= 1.8e-104)
		tmp = t_1;
	elseif (B <= 8.5e-65)
		tmp = (sqrt((F * t_2)) * (B * -sqrt(2.0))) / t_0;
	elseif (B <= 7.2e-45)
		tmp = t_1;
	else
		tmp = -sqrt((F * (2.0 * t_2))) / B;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
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[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-N[Sqrt[N[(2.0 * N[(N[(t$95$0 * F), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(A - N[Sqrt[A ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, 1.8e-104], t$95$1, If[LessEqual[B, 8.5e-65], N[(N[(N[Sqrt[N[(F * t$95$2), $MachinePrecision]], $MachinePrecision] * N[(B * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[B, 7.2e-45], t$95$1, N[((-N[Sqrt[N[(F * N[(2.0 * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / B), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.7999999999999999e-104 or 8.5000000000000003e-65 < B < 7.20000000000000001e-45

   1. Initial program 15.2%

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

   3. Simplified 15.2%

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

   4. Taylor expanded in A around -inf 15.2%

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

      1. *-commutative 15.2%

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

   6. Simplified 15.2%

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

   if 1.7999999999999999e-104 < B < 8.5000000000000003e-65

   1. Initial program 22.3%

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

   3. Simplified 22.3%

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

   4. Taylor expanded in C around 0 43.9%

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

      1. *-commutative 43.9%

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

      2. +-commutative 43.9%

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

      3. unpow2 43.9%

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

      4. unpow2 43.9%

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

      5. hypot-def 44.4%

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

      6. *-commutative 44.4%

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

   6. Simplified 44.4%

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

   if 7.20000000000000001e-45 < B

   1. Initial program 15.8%

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

   3. Simplified 15.8%

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

   4. Taylor expanded in C around 0 25.1%

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

      1. mul-1-neg 25.1%

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

      2. *-commutative 25.1%

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

      3. +-commutative 25.1%

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

      4. unpow2 25.1%

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

      5. unpow2 25.1%

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

      6. hypot-def 41.8%

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

   6. Simplified 41.8%

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

      1. associate-*l/ 41.9%

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

   8. Applied egg-rr 41.9%

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

      1. expm1-log1p-u 40.4%

         \[\leadsto -\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}\right)\right)}}{B} \]

      2. expm1-udef 26.8%

         \[\leadsto -\frac{\color{blue}{e^{\mathsf{log1p}\left(\sqrt{2} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}\right)} - 1}}{B} \]

      3. sqrt-unprod 26.8%

         \[\leadsto -\frac{e^{\mathsf{log1p}\left(\color{blue}{\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)\right)}}\right)} - 1}{B} \]

   10. Applied egg-rr 26.8%

       \[\leadsto -\frac{\color{blue}{e^{\mathsf{log1p}\left(\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)\right)}\right)} - 1}}{B} \]
   11. Step-by-step derivation

       1. expm1-def 40.4%

          \[\leadsto -\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)\right)}\right)\right)}}{B} \]

       2. expm1-log1p 42.0%

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

       3. *-commutative 42.0%

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

       4. associate-*l* 42.0%

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

   12. Simplified 42.0%

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

4. Final simplification 23.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.8 \cdot 10^{-104}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(2 \cdot A\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 8.5 \cdot 10^{-65}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)} \cdot \left(B \cdot \left(-\sqrt{2}\right)\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 7.2 \cdot 10^{-45}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(2 \cdot A\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F \cdot \left(2 \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)\right)}}{B}\\ \end{array} \]

Alternative 3: 44.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} B = |B|\\ [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := A - \mathsf{hypot}\left(A, B\right)\\ t_1 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ t_2 := \frac{-\sqrt{2 \cdot \left(\left(t_1 \cdot F\right) \cdot \left(2 \cdot A\right)\right)}}{t_1}\\ \mathbf{if}\;B \leq 2.2 \cdot 10^{-104}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 4.4 \cdot 10^{-65}:\\ \;\;\;\;\sqrt{F \cdot t_0} \cdot \frac{-\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq 5.2 \cdot 10^{-42}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F \cdot \left(2 \cdot t_0\right)}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
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 (hypot A B)))
        (t_1 (- (* B B) (* 4.0 (* A C))))
        (t_2 (/ (- (sqrt (* 2.0 (* (* t_1 F) (* 2.0 A))))) t_1)))
   (if (<= B 2.2e-104)
     t_2
     (if (<= B 4.4e-65)
       (* (sqrt (* F t_0)) (/ (- (sqrt 2.0)) B))
       (if (<= B 5.2e-42) t_2 (/ (- (sqrt (* F (* 2.0 t_0)))) B))))))

C

B = abs(B);
assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = A - hypot(A, B);
	double t_1 = (B * B) - (4.0 * (A * C));
	double t_2 = -sqrt((2.0 * ((t_1 * F) * (2.0 * A)))) / t_1;
	double tmp;
	if (B <= 2.2e-104) {
		tmp = t_2;
	} else if (B <= 4.4e-65) {
		tmp = sqrt((F * t_0)) * (-sqrt(2.0) / B);
	} else if (B <= 5.2e-42) {
		tmp = t_2;
	} else {
		tmp = -sqrt((F * (2.0 * t_0))) / B;
	}
	return tmp;
}

Java

B = Math.abs(B);
assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = A - Math.hypot(A, B);
	double t_1 = (B * B) - (4.0 * (A * C));
	double t_2 = -Math.sqrt((2.0 * ((t_1 * F) * (2.0 * A)))) / t_1;
	double tmp;
	if (B <= 2.2e-104) {
		tmp = t_2;
	} else if (B <= 4.4e-65) {
		tmp = Math.sqrt((F * t_0)) * (-Math.sqrt(2.0) / B);
	} else if (B <= 5.2e-42) {
		tmp = t_2;
	} else {
		tmp = -Math.sqrt((F * (2.0 * t_0))) / B;
	}
	return tmp;
}

Python

B = abs(B)
[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = A - math.hypot(A, B)
	t_1 = (B * B) - (4.0 * (A * C))
	t_2 = -math.sqrt((2.0 * ((t_1 * F) * (2.0 * A)))) / t_1
	tmp = 0
	if B <= 2.2e-104:
		tmp = t_2
	elif B <= 4.4e-65:
		tmp = math.sqrt((F * t_0)) * (-math.sqrt(2.0) / B)
	elif B <= 5.2e-42:
		tmp = t_2
	else:
		tmp = -math.sqrt((F * (2.0 * t_0))) / B
	return tmp

Julia

B = abs(B)
A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(A - hypot(A, B))
	t_1 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	t_2 = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(t_1 * F) * Float64(2.0 * A))))) / t_1)
	tmp = 0.0
	if (B <= 2.2e-104)
		tmp = t_2;
	elseif (B <= 4.4e-65)
		tmp = Float64(sqrt(Float64(F * t_0)) * Float64(Float64(-sqrt(2.0)) / B));
	elseif (B <= 5.2e-42)
		tmp = t_2;
	else
		tmp = Float64(Float64(-sqrt(Float64(F * Float64(2.0 * t_0)))) / B);
	end
	return tmp
end

MATLAB

B = abs(B)
A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = A - hypot(A, B);
	t_1 = (B * B) - (4.0 * (A * C));
	t_2 = -sqrt((2.0 * ((t_1 * F) * (2.0 * A)))) / t_1;
	tmp = 0.0;
	if (B <= 2.2e-104)
		tmp = t_2;
	elseif (B <= 4.4e-65)
		tmp = sqrt((F * t_0)) * (-sqrt(2.0) / B);
	elseif (B <= 5.2e-42)
		tmp = t_2;
	else
		tmp = -sqrt((F * (2.0 * t_0))) / B;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(A - N[Sqrt[A ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-N[Sqrt[N[(2.0 * N[(N[(t$95$1 * F), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision]}, If[LessEqual[B, 2.2e-104], t$95$2, If[LessEqual[B, 4.4e-65], N[(N[Sqrt[N[(F * t$95$0), $MachinePrecision]], $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 5.2e-42], t$95$2, N[((-N[Sqrt[N[(F * N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / B), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if B < 2.20000000000000012e-104 or 4.40000000000000042e-65 < B < 5.2e-42

   1. Initial program 15.2%

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

   3. Simplified 15.2%

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

   4. Taylor expanded in A around -inf 15.2%

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

      1. *-commutative 15.2%

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

   6. Simplified 15.2%

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

   if 2.20000000000000012e-104 < B < 4.40000000000000042e-65

   1. Initial program 22.3%

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

   3. Simplified 22.3%

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

   4. Taylor expanded in C around 0 44.1%

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

      1. mul-1-neg 44.1%

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

      2. *-commutative 44.1%

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

      3. +-commutative 44.1%

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

      4. unpow2 44.1%

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

      5. unpow2 44.1%

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

      6. hypot-def 44.6%

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

   6. Simplified 44.6%

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

   if 5.2e-42 < B

   1. Initial program 15.8%

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

   3. Simplified 15.8%

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

   4. Taylor expanded in C around 0 25.1%

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

      1. mul-1-neg 25.1%

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

      2. *-commutative 25.1%

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

      3. +-commutative 25.1%

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

      4. unpow2 25.1%

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

      5. unpow2 25.1%

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

      6. hypot-def 41.8%

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

   6. Simplified 41.8%

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

      1. associate-*l/ 41.9%

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

   8. Applied egg-rr 41.9%

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

      1. expm1-log1p-u 40.4%

         \[\leadsto -\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}\right)\right)}}{B} \]

      2. expm1-udef 26.8%

         \[\leadsto -\frac{\color{blue}{e^{\mathsf{log1p}\left(\sqrt{2} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}\right)} - 1}}{B} \]

      3. sqrt-unprod 26.8%

         \[\leadsto -\frac{e^{\mathsf{log1p}\left(\color{blue}{\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)\right)}}\right)} - 1}{B} \]

   10. Applied egg-rr 26.8%

       \[\leadsto -\frac{\color{blue}{e^{\mathsf{log1p}\left(\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)\right)}\right)} - 1}}{B} \]
   11. Step-by-step derivation

       1. expm1-def 40.4%

          \[\leadsto -\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)\right)}\right)\right)}}{B} \]

       2. expm1-log1p 42.0%

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

       3. *-commutative 42.0%

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

       4. associate-*l* 42.0%

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

   12. Simplified 42.0%

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

4. Final simplification 23.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.2 \cdot 10^{-104}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(2 \cdot A\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 4.4 \cdot 10^{-65}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq 5.2 \cdot 10^{-42}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(2 \cdot A\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F \cdot \left(2 \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)\right)}}{B}\\ \end{array} \]

Alternative 4: 40.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} B = |B|\\ [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := \sqrt{B \cdot \left(-F\right)} \cdot \frac{-\sqrt{2}}{B}\\ t_1 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ t_2 := t_1 \cdot F\\ t_3 := \frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(2 \cdot A\right)\right)}}{t_1}\\ t_4 := B \cdot B + \left(A \cdot C\right) \cdot -4\\ \mathbf{if}\;B \leq 9.4 \cdot 10^{-104}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 4.4 \cdot 10^{-65}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 4.5 \cdot 10^{-42}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 2.6 \cdot 10^{-23}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 31000000:\\ \;\;\;\;\sqrt{2 \cdot \left(t_4 \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)} \cdot \frac{-1}{t_4}\\ \mathbf{elif}\;B \leq 5 \cdot 10^{+47}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(\left(A + C\right) - B\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
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 (* B (- F))) (/ (- (sqrt 2.0)) B)))
        (t_1 (- (* B B) (* 4.0 (* A C))))
        (t_2 (* t_1 F))
        (t_3 (/ (- (sqrt (* 2.0 (* t_2 (* 2.0 A))))) t_1))
        (t_4 (+ (* B B) (* (* A C) -4.0))))
   (if (<= B 9.4e-104)
     t_3
     (if (<= B 4.4e-65)
       t_0
       (if (<= B 4.5e-42)
         t_3
         (if (<= B 2.6e-23)
           t_0
           (if (<= B 31000000.0)
             (* (sqrt (* 2.0 (* t_4 (* F (* 2.0 A))))) (/ -1.0 t_4))
             (if (<= B 5e+47)
               (/ (- (sqrt (* 2.0 (* t_2 (- (+ A C) B))))) t_1)
               t_0))))))))

C

B = abs(B);
assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = sqrt((B * -F)) * (-sqrt(2.0) / B);
	double t_1 = (B * B) - (4.0 * (A * C));
	double t_2 = t_1 * F;
	double t_3 = -sqrt((2.0 * (t_2 * (2.0 * A)))) / t_1;
	double t_4 = (B * B) + ((A * C) * -4.0);
	double tmp;
	if (B <= 9.4e-104) {
		tmp = t_3;
	} else if (B <= 4.4e-65) {
		tmp = t_0;
	} else if (B <= 4.5e-42) {
		tmp = t_3;
	} else if (B <= 2.6e-23) {
		tmp = t_0;
	} else if (B <= 31000000.0) {
		tmp = sqrt((2.0 * (t_4 * (F * (2.0 * A))))) * (-1.0 / t_4);
	} else if (B <= 5e+47) {
		tmp = -sqrt((2.0 * (t_2 * ((A + C) - B)))) / t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: B should be positive before calling this function
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) :: t_4
    real(8) :: tmp
    t_0 = sqrt((b * -f)) * (-sqrt(2.0d0) / b)
    t_1 = (b * b) - (4.0d0 * (a * c))
    t_2 = t_1 * f
    t_3 = -sqrt((2.0d0 * (t_2 * (2.0d0 * a)))) / t_1
    t_4 = (b * b) + ((a * c) * (-4.0d0))
    if (b <= 9.4d-104) then
        tmp = t_3
    else if (b <= 4.4d-65) then
        tmp = t_0
    else if (b <= 4.5d-42) then
        tmp = t_3
    else if (b <= 2.6d-23) then
        tmp = t_0
    else if (b <= 31000000.0d0) then
        tmp = sqrt((2.0d0 * (t_4 * (f * (2.0d0 * a))))) * ((-1.0d0) / t_4)
    else if (b <= 5d+47) then
        tmp = -sqrt((2.0d0 * (t_2 * ((a + c) - b)))) / t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

B = Math.abs(B);
assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = Math.sqrt((B * -F)) * (-Math.sqrt(2.0) / B);
	double t_1 = (B * B) - (4.0 * (A * C));
	double t_2 = t_1 * F;
	double t_3 = -Math.sqrt((2.0 * (t_2 * (2.0 * A)))) / t_1;
	double t_4 = (B * B) + ((A * C) * -4.0);
	double tmp;
	if (B <= 9.4e-104) {
		tmp = t_3;
	} else if (B <= 4.4e-65) {
		tmp = t_0;
	} else if (B <= 4.5e-42) {
		tmp = t_3;
	} else if (B <= 2.6e-23) {
		tmp = t_0;
	} else if (B <= 31000000.0) {
		tmp = Math.sqrt((2.0 * (t_4 * (F * (2.0 * A))))) * (-1.0 / t_4);
	} else if (B <= 5e+47) {
		tmp = -Math.sqrt((2.0 * (t_2 * ((A + C) - B)))) / t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

B = abs(B)
[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = math.sqrt((B * -F)) * (-math.sqrt(2.0) / B)
	t_1 = (B * B) - (4.0 * (A * C))
	t_2 = t_1 * F
	t_3 = -math.sqrt((2.0 * (t_2 * (2.0 * A)))) / t_1
	t_4 = (B * B) + ((A * C) * -4.0)
	tmp = 0
	if B <= 9.4e-104:
		tmp = t_3
	elif B <= 4.4e-65:
		tmp = t_0
	elif B <= 4.5e-42:
		tmp = t_3
	elif B <= 2.6e-23:
		tmp = t_0
	elif B <= 31000000.0:
		tmp = math.sqrt((2.0 * (t_4 * (F * (2.0 * A))))) * (-1.0 / t_4)
	elif B <= 5e+47:
		tmp = -math.sqrt((2.0 * (t_2 * ((A + C) - B)))) / t_1
	else:
		tmp = t_0
	return tmp

Julia

B = abs(B)
A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(sqrt(Float64(B * Float64(-F))) * Float64(Float64(-sqrt(2.0)) / B))
	t_1 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	t_2 = Float64(t_1 * F)
	t_3 = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_2 * Float64(2.0 * A))))) / t_1)
	t_4 = Float64(Float64(B * B) + Float64(Float64(A * C) * -4.0))
	tmp = 0.0
	if (B <= 9.4e-104)
		tmp = t_3;
	elseif (B <= 4.4e-65)
		tmp = t_0;
	elseif (B <= 4.5e-42)
		tmp = t_3;
	elseif (B <= 2.6e-23)
		tmp = t_0;
	elseif (B <= 31000000.0)
		tmp = Float64(sqrt(Float64(2.0 * Float64(t_4 * Float64(F * Float64(2.0 * A))))) * Float64(-1.0 / t_4));
	elseif (B <= 5e+47)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_2 * Float64(Float64(A + C) - B))))) / t_1);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

B = abs(B)
A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = sqrt((B * -F)) * (-sqrt(2.0) / B);
	t_1 = (B * B) - (4.0 * (A * C));
	t_2 = t_1 * F;
	t_3 = -sqrt((2.0 * (t_2 * (2.0 * A)))) / t_1;
	t_4 = (B * B) + ((A * C) * -4.0);
	tmp = 0.0;
	if (B <= 9.4e-104)
		tmp = t_3;
	elseif (B <= 4.4e-65)
		tmp = t_0;
	elseif (B <= 4.5e-42)
		tmp = t_3;
	elseif (B <= 2.6e-23)
		tmp = t_0;
	elseif (B <= 31000000.0)
		tmp = sqrt((2.0 * (t_4 * (F * (2.0 * A))))) * (-1.0 / t_4);
	elseif (B <= 5e+47)
		tmp = -sqrt((2.0 * (t_2 * ((A + C) - B)))) / t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
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[Sqrt[N[(B * (-F)), $MachinePrecision]], $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * F), $MachinePrecision]}, Block[{t$95$3 = N[((-N[Sqrt[N[(2.0 * N[(t$95$2 * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[(N[(B * B), $MachinePrecision] + N[(N[(A * C), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, 9.4e-104], t$95$3, If[LessEqual[B, 4.4e-65], t$95$0, If[LessEqual[B, 4.5e-42], t$95$3, If[LessEqual[B, 2.6e-23], t$95$0, If[LessEqual[B, 31000000.0], N[(N[Sqrt[N[(2.0 * N[(t$95$4 * N[(F * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / t$95$4), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 5e+47], N[((-N[Sqrt[N[(2.0 * N[(t$95$2 * N[(N[(A + C), $MachinePrecision] - B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], t$95$0]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 9.4e-104 or 4.40000000000000042e-65 < B < 4.5e-42

   1. Initial program 15.7%

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

   3. Simplified 15.7%

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

   4. Taylor expanded in A around -inf 15.1%

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

      1. *-commutative 15.1%

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

   6. Simplified 15.1%

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

   if 9.4e-104 < B < 4.40000000000000042e-65 or 4.5e-42 < B < 2.6e-23 or 5.00000000000000022e47 < B

   1. Initial program 10.9%

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

   3. Simplified 10.9%

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

   4. Taylor expanded in C around 0 25.2%

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

      1. mul-1-neg 25.2%

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

      2. *-commutative 25.2%

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

      3. +-commutative 25.2%

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

      4. unpow2 25.2%

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

      5. unpow2 25.2%

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

      6. hypot-def 44.2%

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

   6. Simplified 44.2%

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

   7. Taylor expanded in A around 0 38.6%

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

      1. mul-1-neg 38.6%

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

   9. Simplified 38.6%

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

   if 2.6e-23 < B < 3.1e7

   1. Initial program 16.1%

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

   3. Simplified 16.1%

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

   4. Taylor expanded in A around -inf 17.6%

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

      1. *-commutative 17.6%

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

   6. Simplified 17.6%

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

      1. div-inv 17.6%

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

      2. associate-*l* 17.6%

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

      3. cancel-sign-sub-inv 17.6%

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

      4. metadata-eval 17.6%

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

      5. cancel-sign-sub-inv 17.6%

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

      6. metadata-eval 17.6%

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

   8. Applied egg-rr 17.6%

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

   if 3.1e7 < B < 5.00000000000000022e47

   1. Initial program 38.2%

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

   3. Simplified 38.2%

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

   4. Taylor expanded in B around inf 40.6%

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

4. Final simplification 22.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 9.4 \cdot 10^{-104}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(2 \cdot A\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 4.4 \cdot 10^{-65}:\\ \;\;\;\;\sqrt{B \cdot \left(-F\right)} \cdot \frac{-\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq 4.5 \cdot 10^{-42}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(2 \cdot A\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 2.6 \cdot 10^{-23}:\\ \;\;\;\;\sqrt{B \cdot \left(-F\right)} \cdot \frac{-\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq 31000000:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(B \cdot B + \left(A \cdot C\right) \cdot -4\right) \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)} \cdot \frac{-1}{B \cdot B + \left(A \cdot C\right) \cdot -4}\\ \mathbf{elif}\;B \leq 5 \cdot 10^{+47}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(\left(A + C\right) - B\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{B \cdot \left(-F\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 5: 40.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} B = |B|\\ [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := -\sqrt{2}\\ t_1 := \sqrt{B \cdot \left(-F\right)}\\ t_2 := t_1 \cdot \frac{t_0}{B}\\ t_3 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ t_4 := t_3 \cdot F\\ t_5 := \frac{-\sqrt{2 \cdot \left(t_4 \cdot \left(2 \cdot A\right)\right)}}{t_3}\\ t_6 := B \cdot B + \left(A \cdot C\right) \cdot -4\\ \mathbf{if}\;B \leq 9 \cdot 10^{-104}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;B \leq 6.6 \cdot 10^{-65}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 2.6 \cdot 10^{-41}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;B \leq 1.7 \cdot 10^{-24}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 27000000:\\ \;\;\;\;\sqrt{2 \cdot \left(t_6 \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)} \cdot \frac{-1}{t_6}\\ \mathbf{elif}\;B \leq 5.2 \cdot 10^{+47}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_4 \cdot \left(\left(A + C\right) - B\right)\right)}}{t_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1 \cdot t_0}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
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 2.0)))
        (t_1 (sqrt (* B (- F))))
        (t_2 (* t_1 (/ t_0 B)))
        (t_3 (- (* B B) (* 4.0 (* A C))))
        (t_4 (* t_3 F))
        (t_5 (/ (- (sqrt (* 2.0 (* t_4 (* 2.0 A))))) t_3))
        (t_6 (+ (* B B) (* (* A C) -4.0))))
   (if (<= B 9e-104)
     t_5
     (if (<= B 6.6e-65)
       t_2
       (if (<= B 2.6e-41)
         t_5
         (if (<= B 1.7e-24)
           t_2
           (if (<= B 27000000.0)
             (* (sqrt (* 2.0 (* t_6 (* F (* 2.0 A))))) (/ -1.0 t_6))
             (if (<= B 5.2e+47)
               (/ (- (sqrt (* 2.0 (* t_4 (- (+ A C) B))))) t_3)
               (/ (* t_1 t_0) B)))))))))

C

B = abs(B);
assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = -sqrt(2.0);
	double t_1 = sqrt((B * -F));
	double t_2 = t_1 * (t_0 / B);
	double t_3 = (B * B) - (4.0 * (A * C));
	double t_4 = t_3 * F;
	double t_5 = -sqrt((2.0 * (t_4 * (2.0 * A)))) / t_3;
	double t_6 = (B * B) + ((A * C) * -4.0);
	double tmp;
	if (B <= 9e-104) {
		tmp = t_5;
	} else if (B <= 6.6e-65) {
		tmp = t_2;
	} else if (B <= 2.6e-41) {
		tmp = t_5;
	} else if (B <= 1.7e-24) {
		tmp = t_2;
	} else if (B <= 27000000.0) {
		tmp = sqrt((2.0 * (t_6 * (F * (2.0 * A))))) * (-1.0 / t_6);
	} else if (B <= 5.2e+47) {
		tmp = -sqrt((2.0 * (t_4 * ((A + C) - B)))) / t_3;
	} else {
		tmp = (t_1 * t_0) / B;
	}
	return tmp;
}

Fortran

NOTE: B should be positive before calling this function
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) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: tmp
    t_0 = -sqrt(2.0d0)
    t_1 = sqrt((b * -f))
    t_2 = t_1 * (t_0 / b)
    t_3 = (b * b) - (4.0d0 * (a * c))
    t_4 = t_3 * f
    t_5 = -sqrt((2.0d0 * (t_4 * (2.0d0 * a)))) / t_3
    t_6 = (b * b) + ((a * c) * (-4.0d0))
    if (b <= 9d-104) then
        tmp = t_5
    else if (b <= 6.6d-65) then
        tmp = t_2
    else if (b <= 2.6d-41) then
        tmp = t_5
    else if (b <= 1.7d-24) then
        tmp = t_2
    else if (b <= 27000000.0d0) then
        tmp = sqrt((2.0d0 * (t_6 * (f * (2.0d0 * a))))) * ((-1.0d0) / t_6)
    else if (b <= 5.2d+47) then
        tmp = -sqrt((2.0d0 * (t_4 * ((a + c) - b)))) / t_3
    else
        tmp = (t_1 * t_0) / b
    end if
    code = tmp
end function

Java

B = Math.abs(B);
assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = -Math.sqrt(2.0);
	double t_1 = Math.sqrt((B * -F));
	double t_2 = t_1 * (t_0 / B);
	double t_3 = (B * B) - (4.0 * (A * C));
	double t_4 = t_3 * F;
	double t_5 = -Math.sqrt((2.0 * (t_4 * (2.0 * A)))) / t_3;
	double t_6 = (B * B) + ((A * C) * -4.0);
	double tmp;
	if (B <= 9e-104) {
		tmp = t_5;
	} else if (B <= 6.6e-65) {
		tmp = t_2;
	} else if (B <= 2.6e-41) {
		tmp = t_5;
	} else if (B <= 1.7e-24) {
		tmp = t_2;
	} else if (B <= 27000000.0) {
		tmp = Math.sqrt((2.0 * (t_6 * (F * (2.0 * A))))) * (-1.0 / t_6);
	} else if (B <= 5.2e+47) {
		tmp = -Math.sqrt((2.0 * (t_4 * ((A + C) - B)))) / t_3;
	} else {
		tmp = (t_1 * t_0) / B;
	}
	return tmp;
}

Python

B = abs(B)
[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = -math.sqrt(2.0)
	t_1 = math.sqrt((B * -F))
	t_2 = t_1 * (t_0 / B)
	t_3 = (B * B) - (4.0 * (A * C))
	t_4 = t_3 * F
	t_5 = -math.sqrt((2.0 * (t_4 * (2.0 * A)))) / t_3
	t_6 = (B * B) + ((A * C) * -4.0)
	tmp = 0
	if B <= 9e-104:
		tmp = t_5
	elif B <= 6.6e-65:
		tmp = t_2
	elif B <= 2.6e-41:
		tmp = t_5
	elif B <= 1.7e-24:
		tmp = t_2
	elif B <= 27000000.0:
		tmp = math.sqrt((2.0 * (t_6 * (F * (2.0 * A))))) * (-1.0 / t_6)
	elif B <= 5.2e+47:
		tmp = -math.sqrt((2.0 * (t_4 * ((A + C) - B)))) / t_3
	else:
		tmp = (t_1 * t_0) / B
	return tmp

Julia

B = abs(B)
A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(-sqrt(2.0))
	t_1 = sqrt(Float64(B * Float64(-F)))
	t_2 = Float64(t_1 * Float64(t_0 / B))
	t_3 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	t_4 = Float64(t_3 * F)
	t_5 = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_4 * Float64(2.0 * A))))) / t_3)
	t_6 = Float64(Float64(B * B) + Float64(Float64(A * C) * -4.0))
	tmp = 0.0
	if (B <= 9e-104)
		tmp = t_5;
	elseif (B <= 6.6e-65)
		tmp = t_2;
	elseif (B <= 2.6e-41)
		tmp = t_5;
	elseif (B <= 1.7e-24)
		tmp = t_2;
	elseif (B <= 27000000.0)
		tmp = Float64(sqrt(Float64(2.0 * Float64(t_6 * Float64(F * Float64(2.0 * A))))) * Float64(-1.0 / t_6));
	elseif (B <= 5.2e+47)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_4 * Float64(Float64(A + C) - B))))) / t_3);
	else
		tmp = Float64(Float64(t_1 * t_0) / B);
	end
	return tmp
end

MATLAB

B = abs(B)
A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = -sqrt(2.0);
	t_1 = sqrt((B * -F));
	t_2 = t_1 * (t_0 / B);
	t_3 = (B * B) - (4.0 * (A * C));
	t_4 = t_3 * F;
	t_5 = -sqrt((2.0 * (t_4 * (2.0 * A)))) / t_3;
	t_6 = (B * B) + ((A * C) * -4.0);
	tmp = 0.0;
	if (B <= 9e-104)
		tmp = t_5;
	elseif (B <= 6.6e-65)
		tmp = t_2;
	elseif (B <= 2.6e-41)
		tmp = t_5;
	elseif (B <= 1.7e-24)
		tmp = t_2;
	elseif (B <= 27000000.0)
		tmp = sqrt((2.0 * (t_6 * (F * (2.0 * A))))) * (-1.0 / t_6);
	elseif (B <= 5.2e+47)
		tmp = -sqrt((2.0 * (t_4 * ((A + C) - B)))) / t_3;
	else
		tmp = (t_1 * t_0) / B;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
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[2.0], $MachinePrecision])}, Block[{t$95$1 = N[Sqrt[N[(B * (-F)), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(t$95$0 / B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * F), $MachinePrecision]}, Block[{t$95$5 = N[((-N[Sqrt[N[(2.0 * N[(t$95$4 * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$3), $MachinePrecision]}, Block[{t$95$6 = N[(N[(B * B), $MachinePrecision] + N[(N[(A * C), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, 9e-104], t$95$5, If[LessEqual[B, 6.6e-65], t$95$2, If[LessEqual[B, 2.6e-41], t$95$5, If[LessEqual[B, 1.7e-24], t$95$2, If[LessEqual[B, 27000000.0], N[(N[Sqrt[N[(2.0 * N[(t$95$6 * N[(F * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / t$95$6), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 5.2e+47], N[((-N[Sqrt[N[(2.0 * N[(t$95$4 * N[(N[(A + C), $MachinePrecision] - B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$3), $MachinePrecision], N[(N[(t$95$1 * t$95$0), $MachinePrecision] / B), $MachinePrecision]]]]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if B < 8.9999999999999995e-104 or 6.6000000000000002e-65 < B < 2.5999999999999999e-41

   1. Initial program 15.7%

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

   3. Simplified 15.7%

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

   4. Taylor expanded in A around -inf 15.1%

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

      1. *-commutative 15.1%

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

   6. Simplified 15.1%

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

   if 8.9999999999999995e-104 < B < 6.6000000000000002e-65 or 2.5999999999999999e-41 < B < 1.69999999999999996e-24

   1. Initial program 27.4%

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

   3. Simplified 27.4%

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

   4. Taylor expanded in C around 0 63.8%

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

      1. mul-1-neg 63.8%

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

      2. *-commutative 63.8%

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

      3. +-commutative 63.8%

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

      4. unpow2 63.8%

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

      5. unpow2 63.8%

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

      6. hypot-def 64.1%

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

   6. Simplified 64.1%

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

   7. Taylor expanded in A around 0 64.2%

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

      1. mul-1-neg 64.2%

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

   9. Simplified 64.2%

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

   if 1.69999999999999996e-24 < B < 2.7e7

   1. Initial program 16.1%

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

   3. Simplified 16.1%

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

   4. Taylor expanded in A around -inf 17.6%

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

      1. *-commutative 17.6%

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

   6. Simplified 17.6%

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

      1. div-inv 17.6%

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

      2. associate-*l* 17.6%

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

      3. cancel-sign-sub-inv 17.6%

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

      4. metadata-eval 17.6%

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

      5. cancel-sign-sub-inv 17.6%

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

      6. metadata-eval 17.6%

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

   8. Applied egg-rr 17.6%

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

   if 2.7e7 < B < 5.20000000000000007e47

   1. Initial program 38.2%

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

   3. Simplified 38.2%

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

   4. Taylor expanded in B around inf 40.6%

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

   if 5.20000000000000007e47 < B

   1. Initial program 8.5%

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

   3. Simplified 8.5%

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

   4. Taylor expanded in C around 0 19.5%

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

      1. mul-1-neg 19.5%

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

      2. *-commutative 19.5%

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

      3. +-commutative 19.5%

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

      4. unpow2 19.5%

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

      5. unpow2 19.5%

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

      6. hypot-def 41.3%

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

   6. Simplified 41.3%

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

      1. associate-*l/ 41.3%

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

   8. Applied egg-rr 41.3%

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

   9. Taylor expanded in A around 0 34.9%

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

       1. mul-1-neg 34.9%

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

   11. Simplified 34.9%

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

4. Final simplification 22.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 9 \cdot 10^{-104}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(2 \cdot A\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 6.6 \cdot 10^{-65}:\\ \;\;\;\;\sqrt{B \cdot \left(-F\right)} \cdot \frac{-\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq 2.6 \cdot 10^{-41}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(2 \cdot A\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 1.7 \cdot 10^{-24}:\\ \;\;\;\;\sqrt{B \cdot \left(-F\right)} \cdot \frac{-\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq 27000000:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(B \cdot B + \left(A \cdot C\right) \cdot -4\right) \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)} \cdot \frac{-1}{B \cdot B + \left(A \cdot C\right) \cdot -4}\\ \mathbf{elif}\;B \leq 5.2 \cdot 10^{+47}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(\left(A + C\right) - B\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{B \cdot \left(-F\right)} \cdot \left(-\sqrt{2}\right)}{B}\\ \end{array} \]

Alternative 6: 44.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} B = |B|\\ [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq 2.3 \cdot 10^{-104} \lor \neg \left(B \leq 3.2 \cdot 10^{-64}\right) \land B \leq 4.3 \cdot 10^{-44}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(t_0 \cdot F\right) \cdot \left(2 \cdot A\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F \cdot \left(2 \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)\right)}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
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) (* 4.0 (* A C)))))
   (if (or (<= B 2.3e-104) (and (not (<= B 3.2e-64)) (<= B 4.3e-44)))
     (/ (- (sqrt (* 2.0 (* (* t_0 F) (* 2.0 A))))) t_0)
     (/ (- (sqrt (* F (* 2.0 (- A (hypot A B)))))) B))))

C

B = abs(B);
assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double tmp;
	if ((B <= 2.3e-104) || (!(B <= 3.2e-64) && (B <= 4.3e-44))) {
		tmp = -sqrt((2.0 * ((t_0 * F) * (2.0 * A)))) / t_0;
	} else {
		tmp = -sqrt((F * (2.0 * (A - hypot(A, B))))) / B;
	}
	return tmp;
}

Java

B = Math.abs(B);
assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double tmp;
	if ((B <= 2.3e-104) || (!(B <= 3.2e-64) && (B <= 4.3e-44))) {
		tmp = -Math.sqrt((2.0 * ((t_0 * F) * (2.0 * A)))) / t_0;
	} else {
		tmp = -Math.sqrt((F * (2.0 * (A - Math.hypot(A, B))))) / B;
	}
	return tmp;
}

Python

B = abs(B)
[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (A * C))
	tmp = 0
	if (B <= 2.3e-104) or (not (B <= 3.2e-64) and (B <= 4.3e-44)):
		tmp = -math.sqrt((2.0 * ((t_0 * F) * (2.0 * A)))) / t_0
	else:
		tmp = -math.sqrt((F * (2.0 * (A - math.hypot(A, B))))) / B
	return tmp

Julia

B = abs(B)
A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	tmp = 0.0
	if ((B <= 2.3e-104) || (!(B <= 3.2e-64) && (B <= 4.3e-44)))
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(t_0 * F) * Float64(2.0 * A))))) / t_0);
	else
		tmp = Float64(Float64(-sqrt(Float64(F * Float64(2.0 * Float64(A - hypot(A, B)))))) / B);
	end
	return tmp
end

MATLAB

B = abs(B)
A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (A * C));
	tmp = 0.0;
	if ((B <= 2.3e-104) || (~((B <= 3.2e-64)) && (B <= 4.3e-44)))
		tmp = -sqrt((2.0 * ((t_0 * F) * (2.0 * A)))) / t_0;
	else
		tmp = -sqrt((F * (2.0 * (A - hypot(A, B))))) / B;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
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[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[B, 2.3e-104], And[N[Not[LessEqual[B, 3.2e-64]], $MachinePrecision], LessEqual[B, 4.3e-44]]], N[((-N[Sqrt[N[(2.0 * N[(N[(t$95$0 * F), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[((-N[Sqrt[N[(F * N[(2.0 * N[(A - N[Sqrt[A ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / B), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 2.2999999999999999e-104 or 3.19999999999999975e-64 < B < 4.30000000000000013e-44

   1. Initial program 15.2%

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

   3. Simplified 15.2%

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

   4. Taylor expanded in A around -inf 15.2%

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

      1. *-commutative 15.2%

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

   6. Simplified 15.2%

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

   if 2.2999999999999999e-104 < B < 3.19999999999999975e-64 or 4.30000000000000013e-44 < B

   1. Initial program 16.2%

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

   3. Simplified 16.2%

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

   4. Taylor expanded in C around 0 26.2%

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

      1. mul-1-neg 26.2%

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

      2. *-commutative 26.2%

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

      3. +-commutative 26.2%

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

      4. unpow2 26.2%

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

      5. unpow2 26.2%

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

      6. hypot-def 42.0%

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

   6. Simplified 42.0%

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

      1. associate-*l/ 42.1%

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

   8. Applied egg-rr 42.1%

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

      1. expm1-log1p-u 40.6%

         \[\leadsto -\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}\right)\right)}}{B} \]

      2. expm1-udef 25.6%

         \[\leadsto -\frac{\color{blue}{e^{\mathsf{log1p}\left(\sqrt{2} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}\right)} - 1}}{B} \]

      3. sqrt-unprod 25.6%

         \[\leadsto -\frac{e^{\mathsf{log1p}\left(\color{blue}{\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)\right)}}\right)} - 1}{B} \]

   10. Applied egg-rr 25.6%

       \[\leadsto -\frac{\color{blue}{e^{\mathsf{log1p}\left(\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)\right)}\right)} - 1}}{B} \]
   11. Step-by-step derivation

       1. expm1-def 40.6%

          \[\leadsto -\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)\right)}\right)\right)}}{B} \]

       2. expm1-log1p 42.1%

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

       3. *-commutative 42.1%

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

       4. associate-*l* 42.1%

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

   12. Simplified 42.1%

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

4. Final simplification 23.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.3 \cdot 10^{-104} \lor \neg \left(B \leq 3.2 \cdot 10^{-64}\right) \land B \leq 4.3 \cdot 10^{-44}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(2 \cdot A\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F \cdot \left(2 \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)\right)}}{B}\\ \end{array} \]

Alternative 7: 40.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} B = |B|\\ [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ t_1 := \frac{-\sqrt{2 \cdot \left(\left(t_0 \cdot F\right) \cdot \left(2 \cdot A\right)\right)}}{t_0}\\ \mathbf{if}\;B \leq 9.4 \cdot 10^{-104}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 7 \cdot 10^{-65}:\\ \;\;\;\;\sqrt{B \cdot \left(-F\right)} \cdot \frac{-\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq 8.5 \cdot 10^{-43}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - B\right)}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
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) (* 4.0 (* A C))))
        (t_1 (/ (- (sqrt (* 2.0 (* (* t_0 F) (* 2.0 A))))) t_0)))
   (if (<= B 9.4e-104)
     t_1
     (if (<= B 7e-65)
       (* (sqrt (* B (- F))) (/ (- (sqrt 2.0)) B))
       (if (<= B 8.5e-43)
         t_1
         (* (/ (sqrt 2.0) B) (- (sqrt (* F (- A B))))))))))

C

B = abs(B);
assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double t_1 = -sqrt((2.0 * ((t_0 * F) * (2.0 * A)))) / t_0;
	double tmp;
	if (B <= 9.4e-104) {
		tmp = t_1;
	} else if (B <= 7e-65) {
		tmp = sqrt((B * -F)) * (-sqrt(2.0) / B);
	} else if (B <= 8.5e-43) {
		tmp = t_1;
	} else {
		tmp = (sqrt(2.0) / B) * -sqrt((F * (A - B)));
	}
	return tmp;
}

Fortran

NOTE: B should be positive before calling this function
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 = (b * b) - (4.0d0 * (a * c))
    t_1 = -sqrt((2.0d0 * ((t_0 * f) * (2.0d0 * a)))) / t_0
    if (b <= 9.4d-104) then
        tmp = t_1
    else if (b <= 7d-65) then
        tmp = sqrt((b * -f)) * (-sqrt(2.0d0) / b)
    else if (b <= 8.5d-43) then
        tmp = t_1
    else
        tmp = (sqrt(2.0d0) / b) * -sqrt((f * (a - b)))
    end if
    code = tmp
end function

Java

B = Math.abs(B);
assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double t_1 = -Math.sqrt((2.0 * ((t_0 * F) * (2.0 * A)))) / t_0;
	double tmp;
	if (B <= 9.4e-104) {
		tmp = t_1;
	} else if (B <= 7e-65) {
		tmp = Math.sqrt((B * -F)) * (-Math.sqrt(2.0) / B);
	} else if (B <= 8.5e-43) {
		tmp = t_1;
	} else {
		tmp = (Math.sqrt(2.0) / B) * -Math.sqrt((F * (A - B)));
	}
	return tmp;
}

Python

B = abs(B)
[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (A * C))
	t_1 = -math.sqrt((2.0 * ((t_0 * F) * (2.0 * A)))) / t_0
	tmp = 0
	if B <= 9.4e-104:
		tmp = t_1
	elif B <= 7e-65:
		tmp = math.sqrt((B * -F)) * (-math.sqrt(2.0) / B)
	elif B <= 8.5e-43:
		tmp = t_1
	else:
		tmp = (math.sqrt(2.0) / B) * -math.sqrt((F * (A - B)))
	return tmp

Julia

B = abs(B)
A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	t_1 = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(t_0 * F) * Float64(2.0 * A))))) / t_0)
	tmp = 0.0
	if (B <= 9.4e-104)
		tmp = t_1;
	elseif (B <= 7e-65)
		tmp = Float64(sqrt(Float64(B * Float64(-F))) * Float64(Float64(-sqrt(2.0)) / B));
	elseif (B <= 8.5e-43)
		tmp = t_1;
	else
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(-sqrt(Float64(F * Float64(A - B)))));
	end
	return tmp
end

MATLAB

B = abs(B)
A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (A * C));
	t_1 = -sqrt((2.0 * ((t_0 * F) * (2.0 * A)))) / t_0;
	tmp = 0.0;
	if (B <= 9.4e-104)
		tmp = t_1;
	elseif (B <= 7e-65)
		tmp = sqrt((B * -F)) * (-sqrt(2.0) / B);
	elseif (B <= 8.5e-43)
		tmp = t_1;
	else
		tmp = (sqrt(2.0) / B) * -sqrt((F * (A - B)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
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[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-N[Sqrt[N[(2.0 * N[(N[(t$95$0 * F), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision]}, If[LessEqual[B, 9.4e-104], t$95$1, If[LessEqual[B, 7e-65], N[(N[Sqrt[N[(B * (-F)), $MachinePrecision]], $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 8.5e-43], t$95$1, N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * (-N[Sqrt[N[(F * N[(A - B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if B < 9.4e-104 or 7.00000000000000009e-65 < B < 8.50000000000000056e-43

   1. Initial program 15.7%

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

   3. Simplified 15.7%

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

   4. Taylor expanded in A around -inf 15.1%

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

      1. *-commutative 15.1%

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

   6. Simplified 15.1%

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

   if 9.4e-104 < B < 7.00000000000000009e-65

   1. Initial program 3.3%

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

   3. Simplified 3.3%

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

   4. Taylor expanded in C around 0 51.1%

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

      1. mul-1-neg 51.1%

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

      2. *-commutative 51.1%

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

      3. +-commutative 51.1%

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

      4. unpow2 51.1%

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

      5. unpow2 51.1%

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

      6. hypot-def 51.6%

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

   6. Simplified 51.6%

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

   7. Taylor expanded in A around 0 51.8%

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

      1. mul-1-neg 51.8%

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

   9. Simplified 51.8%

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

   if 8.50000000000000056e-43 < B

   1. Initial program 15.8%

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

   3. Simplified 15.8%

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

   4. Taylor expanded in C around 0 25.1%

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

      1. mul-1-neg 25.1%

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

      2. *-commutative 25.1%

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

      3. +-commutative 25.1%

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

      4. unpow2 25.1%

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

      5. unpow2 25.1%

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

      6. hypot-def 41.8%

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

   6. Simplified 41.8%

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

      1. add-log-exp 2.8%

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

   8. Applied egg-rr 2.8%

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

   9. Taylor expanded in A around 0 35.4%

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

       1. neg-mul-1 35.4%

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

       2. unsub-neg 35.4%

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

   11. Simplified 35.4%

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

4. Final simplification 21.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 9.4 \cdot 10^{-104}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(2 \cdot A\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 7 \cdot 10^{-65}:\\ \;\;\;\;\sqrt{B \cdot \left(-F\right)} \cdot \frac{-\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq 8.5 \cdot 10^{-43}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(2 \cdot A\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - B\right)}\right)\\ \end{array} \]

Alternative 8: 29.6% accurate, 4.7× speedup

Math

\[\begin{array}{l} B = |B|\\ [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ t_1 := t_0 \cdot F\\ \mathbf{if}\;B \leq 8 \cdot 10^{-41}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(2 \cdot A\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 8.5 \cdot 10^{+47}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(\left(A + C\right) - B\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{F}{\frac{B}{\frac{A}{B}}}}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
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) (* 4.0 (* A C)))) (t_1 (* t_0 F)))
   (if (<= B 8e-41)
     (/ (- (sqrt (* 2.0 (* t_1 (* 2.0 A))))) t_0)
     (if (<= B 8.5e+47)
       (/ (- (sqrt (* 2.0 (* t_1 (- (+ A C) B))))) t_0)
       (* -2.0 (sqrt (/ F (/ B (/ A B)))))))))

C

B = abs(B);
assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double t_1 = t_0 * F;
	double tmp;
	if (B <= 8e-41) {
		tmp = -sqrt((2.0 * (t_1 * (2.0 * A)))) / t_0;
	} else if (B <= 8.5e+47) {
		tmp = -sqrt((2.0 * (t_1 * ((A + C) - B)))) / t_0;
	} else {
		tmp = -2.0 * sqrt((F / (B / (A / B))));
	}
	return tmp;
}

Fortran

NOTE: B should be positive before calling this function
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 = (b * b) - (4.0d0 * (a * c))
    t_1 = t_0 * f
    if (b <= 8d-41) then
        tmp = -sqrt((2.0d0 * (t_1 * (2.0d0 * a)))) / t_0
    else if (b <= 8.5d+47) then
        tmp = -sqrt((2.0d0 * (t_1 * ((a + c) - b)))) / t_0
    else
        tmp = (-2.0d0) * sqrt((f / (b / (a / b))))
    end if
    code = tmp
end function

Java

B = Math.abs(B);
assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double t_1 = t_0 * F;
	double tmp;
	if (B <= 8e-41) {
		tmp = -Math.sqrt((2.0 * (t_1 * (2.0 * A)))) / t_0;
	} else if (B <= 8.5e+47) {
		tmp = -Math.sqrt((2.0 * (t_1 * ((A + C) - B)))) / t_0;
	} else {
		tmp = -2.0 * Math.sqrt((F / (B / (A / B))));
	}
	return tmp;
}

Python

B = abs(B)
[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (A * C))
	t_1 = t_0 * F
	tmp = 0
	if B <= 8e-41:
		tmp = -math.sqrt((2.0 * (t_1 * (2.0 * A)))) / t_0
	elif B <= 8.5e+47:
		tmp = -math.sqrt((2.0 * (t_1 * ((A + C) - B)))) / t_0
	else:
		tmp = -2.0 * math.sqrt((F / (B / (A / B))))
	return tmp

Julia

B = abs(B)
A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	t_1 = Float64(t_0 * F)
	tmp = 0.0
	if (B <= 8e-41)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_1 * Float64(2.0 * A))))) / t_0);
	elseif (B <= 8.5e+47)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_1 * Float64(Float64(A + C) - B))))) / t_0);
	else
		tmp = Float64(-2.0 * sqrt(Float64(F / Float64(B / Float64(A / B)))));
	end
	return tmp
end

MATLAB

B = abs(B)
A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (A * C));
	t_1 = t_0 * F;
	tmp = 0.0;
	if (B <= 8e-41)
		tmp = -sqrt((2.0 * (t_1 * (2.0 * A)))) / t_0;
	elseif (B <= 8.5e+47)
		tmp = -sqrt((2.0 * (t_1 * ((A + C) - B)))) / t_0;
	else
		tmp = -2.0 * sqrt((F / (B / (A / B))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
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[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * F), $MachinePrecision]}, If[LessEqual[B, 8e-41], N[((-N[Sqrt[N[(2.0 * N[(t$95$1 * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[B, 8.5e+47], N[((-N[Sqrt[N[(2.0 * N[(t$95$1 * N[(N[(A + C), $MachinePrecision] - B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(-2.0 * N[Sqrt[N[(F / N[(B / N[(A / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if B < 8.00000000000000005e-41

   1. Initial program 15.4%

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

   3. Simplified 15.4%

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

   4. Taylor expanded in A around -inf 14.9%

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

      1. *-commutative 14.9%

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

   6. Simplified 14.9%

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

   if 8.00000000000000005e-41 < B < 8.5000000000000008e47

   1. Initial program 33.6%

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

   3. Simplified 33.6%

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

   4. Taylor expanded in B around inf 30.9%

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

   if 8.5000000000000008e47 < B

   1. Initial program 8.5%

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

   3. Simplified 8.5%

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

   4. Taylor expanded in A around -inf 4.7%

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

      1. *-commutative 4.7%

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

   6. Simplified 4.7%

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

   7. Taylor expanded in B around inf 9.0%

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

      1. *-commutative 9.0%

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

   9. Simplified 9.0%

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

       1. add-sqr-sqrt 9.0%

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

       2. sqrt-unprod 8.5%

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

       3. un-div-inv 8.5%

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

       4. un-div-inv 8.5%

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

       5. frac-times 6.5%

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

       6. add-sqr-sqrt 6.9%

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

   11. Applied egg-rr 6.9%

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

       1. associate-/l* 9.0%

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

       2. associate-/l* 12.6%

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

   13. Simplified 12.6%

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

4. Final simplification 15.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 8 \cdot 10^{-41}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(2 \cdot A\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 8.5 \cdot 10^{+47}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(\left(A + C\right) - B\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{F}{\frac{B}{\frac{A}{B}}}}\\ \end{array} \]

Alternative 9: 30.1% accurate, 4.9× speedup

Math

\[\begin{array}{l} B = |B|\\ [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B + \left(A \cdot C\right) \cdot -4\\ \mathbf{if}\;B \leq 1.4 \cdot 10^{+36}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{F}{\frac{B}{\frac{A}{B}}}}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
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 1.4e+36)
     (/ (- (sqrt (* 2.0 (* t_0 (* F (* 2.0 A)))))) t_0)
     (* -2.0 (sqrt (/ F (/ B (/ A B))))))))

C

B = abs(B);
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 <= 1.4e+36) {
		tmp = -sqrt((2.0 * (t_0 * (F * (2.0 * A))))) / t_0;
	} else {
		tmp = -2.0 * sqrt((F / (B / (A / B))));
	}
	return tmp;
}

Fortran

NOTE: B should be positive before calling this function
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 <= 1.4d+36) then
        tmp = -sqrt((2.0d0 * (t_0 * (f * (2.0d0 * a))))) / t_0
    else
        tmp = (-2.0d0) * sqrt((f / (b / (a / b))))
    end if
    code = tmp
end function

Java

B = Math.abs(B);
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 <= 1.4e+36) {
		tmp = -Math.sqrt((2.0 * (t_0 * (F * (2.0 * A))))) / t_0;
	} else {
		tmp = -2.0 * Math.sqrt((F / (B / (A / B))));
	}
	return tmp;
}

Python

B = abs(B)
[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (B * B) + ((A * C) * -4.0)
	tmp = 0
	if B <= 1.4e+36:
		tmp = -math.sqrt((2.0 * (t_0 * (F * (2.0 * A))))) / t_0
	else:
		tmp = -2.0 * math.sqrt((F / (B / (A / B))))
	return tmp

Julia

B = abs(B)
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 <= 1.4e+36)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_0 * Float64(F * Float64(2.0 * A)))))) / t_0);
	else
		tmp = Float64(-2.0 * sqrt(Float64(F / Float64(B / Float64(A / B)))));
	end
	return tmp
end

MATLAB

B = abs(B)
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 <= 1.4e+36)
		tmp = -sqrt((2.0 * (t_0 * (F * (2.0 * A))))) / t_0;
	else
		tmp = -2.0 * sqrt((F / (B / (A / B))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
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, 1.4e+36], 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], N[(-2.0 * N[Sqrt[N[(F / N[(B / N[(A / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 1.4e36

   1. Initial program 16.7%

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

   3. Simplified 16.7%

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

   4. Taylor expanded in A around -inf 14.9%

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

      1. *-commutative 14.9%

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

   6. Simplified 14.9%

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

      1. distribute-frac-neg 14.9%

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

      2. associate-*l* 14.0%

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

      3. cancel-sign-sub-inv 14.0%

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

      4. metadata-eval 14.0%

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

      5. cancel-sign-sub-inv 14.0%

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

      6. metadata-eval 14.0%

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

   8. Applied egg-rr 14.0%

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

   if 1.4e36 < B

   1. Initial program 11.5%

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

   3. Simplified 11.5%

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

   4. Taylor expanded in A around -inf 6.3%

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

      1. *-commutative 6.3%

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

   6. Simplified 6.3%

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

   7. Taylor expanded in B around inf 10.3%

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

      1. *-commutative 10.3%

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

   9. Simplified 10.3%

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

       1. add-sqr-sqrt 10.3%

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

       2. sqrt-unprod 9.8%

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

       3. un-div-inv 9.8%

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

       4. un-div-inv 9.8%

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

       5. frac-times 7.9%

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

       6. add-sqr-sqrt 8.3%

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

   11. Applied egg-rr 8.3%

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

       1. associate-/l* 10.3%

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

       2. associate-/l* 13.6%

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

   13. Simplified 13.6%

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

4. Final simplification 13.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.4 \cdot 10^{+36}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(B \cdot B + \left(A \cdot C\right) \cdot -4\right) \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{B \cdot B + \left(A \cdot C\right) \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{F}{\frac{B}{\frac{A}{B}}}}\\ \end{array} \]

Alternative 10: 30.1% accurate, 4.9× speedup

Math

\[\begin{array}{l} B = |B|\\ [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq 1.5 \cdot 10^{+36}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(t_0 \cdot F\right) \cdot \left(2 \cdot A\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{F}{\frac{B}{\frac{A}{B}}}}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
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) (* 4.0 (* A C)))))
   (if (<= B 1.5e+36)
     (/ (- (sqrt (* 2.0 (* (* t_0 F) (* 2.0 A))))) t_0)
     (* -2.0 (sqrt (/ F (/ B (/ A B))))))))

C

B = abs(B);
assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (B <= 1.5e+36) {
		tmp = -sqrt((2.0 * ((t_0 * F) * (2.0 * A)))) / t_0;
	} else {
		tmp = -2.0 * sqrt((F / (B / (A / B))));
	}
	return tmp;
}

Fortran

NOTE: B should be positive before calling this function
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) - (4.0d0 * (a * c))
    if (b <= 1.5d+36) then
        tmp = -sqrt((2.0d0 * ((t_0 * f) * (2.0d0 * a)))) / t_0
    else
        tmp = (-2.0d0) * sqrt((f / (b / (a / b))))
    end if
    code = tmp
end function

Java

B = Math.abs(B);
assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (B <= 1.5e+36) {
		tmp = -Math.sqrt((2.0 * ((t_0 * F) * (2.0 * A)))) / t_0;
	} else {
		tmp = -2.0 * Math.sqrt((F / (B / (A / B))));
	}
	return tmp;
}

Python

B = abs(B)
[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (A * C))
	tmp = 0
	if B <= 1.5e+36:
		tmp = -math.sqrt((2.0 * ((t_0 * F) * (2.0 * A)))) / t_0
	else:
		tmp = -2.0 * math.sqrt((F / (B / (A / B))))
	return tmp

Julia

B = abs(B)
A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	tmp = 0.0
	if (B <= 1.5e+36)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(t_0 * F) * Float64(2.0 * A))))) / t_0);
	else
		tmp = Float64(-2.0 * sqrt(Float64(F / Float64(B / Float64(A / B)))));
	end
	return tmp
end

MATLAB

B = abs(B)
A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (A * C));
	tmp = 0.0;
	if (B <= 1.5e+36)
		tmp = -sqrt((2.0 * ((t_0 * F) * (2.0 * A)))) / t_0;
	else
		tmp = -2.0 * sqrt((F / (B / (A / B))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
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[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, 1.5e+36], N[((-N[Sqrt[N[(2.0 * N[(N[(t$95$0 * F), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(-2.0 * N[Sqrt[N[(F / N[(B / N[(A / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 1.5e36

   1. Initial program 16.7%

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

   3. Simplified 16.7%

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

   4. Taylor expanded in A around -inf 14.9%

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

      1. *-commutative 14.9%

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

   6. Simplified 14.9%

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

   if 1.5e36 < B

   1. Initial program 11.5%

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

   3. Simplified 11.5%

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

   4. Taylor expanded in A around -inf 6.3%

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

      1. *-commutative 6.3%

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

   6. Simplified 6.3%

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

   7. Taylor expanded in B around inf 10.3%

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

      1. *-commutative 10.3%

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

   9. Simplified 10.3%

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

       1. add-sqr-sqrt 10.3%

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

       2. sqrt-unprod 9.8%

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

       3. un-div-inv 9.8%

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

       4. un-div-inv 9.8%

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

       5. frac-times 7.9%

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

       6. add-sqr-sqrt 8.3%

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

   11. Applied egg-rr 8.3%

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

       1. associate-/l* 10.3%

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

       2. associate-/l* 13.6%

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

   13. Simplified 13.6%

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

4. Final simplification 14.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.5 \cdot 10^{+36}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(2 \cdot A\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{F}{\frac{B}{\frac{A}{B}}}}\\ \end{array} \]

Alternative 11: 27.1% accurate, 5.1× speedup

Math

\[\begin{array}{l} B = |B|\\ [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} \mathbf{if}\;B \leq 7.4 \cdot 10^{+33}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-8 \cdot \left(A \cdot \left(C \cdot \left(A \cdot F\right)\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{F}{\frac{B}{\frac{A}{B}}}}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (if (<= B 7.4e+33)
   (/
    (- (sqrt (* 2.0 (* -8.0 (* A (* C (* A F)))))))
    (- (* B B) (* 4.0 (* A C))))
   (* -2.0 (sqrt (/ F (/ B (/ A B)))))))

C

B = abs(B);
assert(A < C);
double code(double A, double B, double C, double F) {
	double tmp;
	if (B <= 7.4e+33) {
		tmp = -sqrt((2.0 * (-8.0 * (A * (C * (A * F)))))) / ((B * B) - (4.0 * (A * C)));
	} else {
		tmp = -2.0 * sqrt((F / (B / (A / B))));
	}
	return tmp;
}

Fortran

NOTE: B should be positive before calling this function
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 <= 7.4d+33) then
        tmp = -sqrt((2.0d0 * ((-8.0d0) * (a * (c * (a * f)))))) / ((b * b) - (4.0d0 * (a * c)))
    else
        tmp = (-2.0d0) * sqrt((f / (b / (a / b))))
    end if
    code = tmp
end function

Java

B = Math.abs(B);
assert A < C;
public static double code(double A, double B, double C, double F) {
	double tmp;
	if (B <= 7.4e+33) {
		tmp = -Math.sqrt((2.0 * (-8.0 * (A * (C * (A * F)))))) / ((B * B) - (4.0 * (A * C)));
	} else {
		tmp = -2.0 * Math.sqrt((F / (B / (A / B))));
	}
	return tmp;
}

Python

B = abs(B)
[A, C] = sort([A, C])
def code(A, B, C, F):
	tmp = 0
	if B <= 7.4e+33:
		tmp = -math.sqrt((2.0 * (-8.0 * (A * (C * (A * F)))))) / ((B * B) - (4.0 * (A * C)))
	else:
		tmp = -2.0 * math.sqrt((F / (B / (A / B))))
	return tmp

Julia

B = abs(B)
A, C = sort([A, C])
function code(A, B, C, F)
	tmp = 0.0
	if (B <= 7.4e+33)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(-8.0 * Float64(A * Float64(C * Float64(A * F))))))) / Float64(Float64(B * B) - Float64(4.0 * Float64(A * C))));
	else
		tmp = Float64(-2.0 * sqrt(Float64(F / Float64(B / Float64(A / B)))));
	end
	return tmp
end

MATLAB

B = abs(B)
A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (B <= 7.4e+33)
		tmp = -sqrt((2.0 * (-8.0 * (A * (C * (A * F)))))) / ((B * B) - (4.0 * (A * C)));
	else
		tmp = -2.0 * sqrt((F / (B / (A / B))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := If[LessEqual[B, 7.4e+33], N[((-N[Sqrt[N[(2.0 * N[(-8.0 * N[(A * N[(C * N[(A * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[Sqrt[N[(F / N[(B / N[(A / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 7.3999999999999997e33

   1. Initial program 16.9%

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

   3. Simplified 16.9%

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

   4. Taylor expanded in A around -inf 15.0%

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

      1. *-commutative 15.0%

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

   6. Simplified 15.0%

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

   7. Taylor expanded in B around 0 9.0%

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

      1. unpow2 9.0%

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

      2. *-commutative 9.0%

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

      3. associate-*r* 12.1%

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

      4. associate-*r* 13.7%

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

      5. *-commutative 13.7%

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

   9. Simplified 13.7%

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

   if 7.3999999999999997e33 < B

   1. Initial program 11.2%

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

   3. Simplified 11.2%

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

   4. Taylor expanded in A around -inf 6.3%

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

      1. *-commutative 6.3%

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

   6. Simplified 6.3%

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

   7. Taylor expanded in B around inf 10.2%

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

      1. *-commutative 10.2%

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

   9. Simplified 10.2%

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

       1. add-sqr-sqrt 10.1%

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

       2. sqrt-unprod 9.7%

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

       3. un-div-inv 9.7%

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

       4. un-div-inv 9.7%

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

       5. frac-times 7.8%

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

       6. add-sqr-sqrt 8.2%

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

   11. Applied egg-rr 8.2%

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

       1. associate-/l* 10.1%

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

       2. associate-/l* 13.3%

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

   13. Simplified 13.3%

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

4. Final simplification 13.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 7.4 \cdot 10^{+33}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-8 \cdot \left(A \cdot \left(C \cdot \left(A \cdot F\right)\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{F}{\frac{B}{\frac{A}{B}}}}\\ \end{array} \]

Alternative 12: 9.3% accurate, 5.8× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: B should be positive before calling this function
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(((a / b) * (f / b)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 15.5%

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

3. Simplified 15.5%

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

4. Taylor expanded in A around -inf 12.9%

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

   1. *-commutative 12.9%

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

6. Simplified 12.9%

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

7. Taylor expanded in B around inf 4.3%

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

   1. *-commutative 4.3%

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

9. Simplified 4.3%

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

    1. add-sqr-sqrt 3.9%

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

    2. sqrt-unprod 5.1%

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

    3. un-div-inv 5.1%

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

    4. un-div-inv 5.1%

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

    5. frac-times 3.8%

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

    6. add-sqr-sqrt 4.0%

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

11. Applied egg-rr 4.0%

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

    1. times-frac 6.8%

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

13. Simplified 6.8%

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

14. Final simplification 6.8%

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

Alternative 13: 9.7% accurate, 5.8× speedup

Math

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

FPCore

NOTE: B should be positive before calling this function
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 (/ B (/ A B))))))

C

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

Fortran

NOTE: B should be positive before calling this function
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 / (b / (a / b))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 15.5%

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

3. Simplified 15.5%

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

4. Taylor expanded in A around -inf 12.9%

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

   1. *-commutative 12.9%

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

6. Simplified 12.9%

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

7. Taylor expanded in B around inf 4.3%

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

   1. *-commutative 4.3%

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

9. Simplified 4.3%

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

    1. add-sqr-sqrt 3.9%

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

    2. sqrt-unprod 5.1%

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

    3. un-div-inv 5.1%

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

    4. un-div-inv 5.1%

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

    5. frac-times 3.8%

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

    6. add-sqr-sqrt 4.0%

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

11. Applied egg-rr 4.0%

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

    1. associate-/l* 5.5%

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

    2. associate-/l* 7.1%

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

13. Simplified 7.1%

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

14. Final simplification 7.1%

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

Alternative 14: 8.8% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: B should be positive before calling this function
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((a * f)) / b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: B should be positive before calling this function
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[(A * F), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 15.5%

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

3. Simplified 15.5%

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

4. Taylor expanded in A around -inf 12.9%

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

   1. *-commutative 12.9%

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

6. Simplified 12.9%

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

7. Taylor expanded in B around inf 4.3%

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

   1. *-commutative 4.3%

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

9. Simplified 4.3%

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

10. Taylor expanded in B around 0 4.3%

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

    1. associate-*r/ 4.3%

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

    2. *-commutative 4.3%

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

    3. *-rgt-identity 4.3%

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

12. Simplified 4.3%

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

13. Final simplification 4.3%

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

Reproduce

herbie shell --seed 2023229 
(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))))