ABCF->ab-angle a

Percentage Accurate: 19.1% → 54.7%

Time: 26.2s

Alternatives: 18

Speedup: 4.5×

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: 54.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ t_1 := \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\\ t_2 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{-\sqrt{\left(2 \cdot \left(t_2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{t_2}\\ t_4 := \frac{\sqrt{2 \cdot \left(F \cdot t_0\right)} \cdot \left(-\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}\right)}{t_0}\\ \mathbf{if}\;t_3 \leq -1 \cdot 10^{-204}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t_3 \leq 2 \cdot 10^{+17}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_1\right) \cdot \mathsf{fma}\left(2, C, -0.5 \cdot \frac{B \cdot B}{A}\right)\right)}}{t_1}\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{A + \mathsf{hypot}\left(B, A\right)}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* A C))))
        (t_1 (fma C (* A -4.0) (* B B)))
        (t_2 (- (pow B 2.0) (* (* 4.0 A) C)))
        (t_3
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_2 F))
             (+ (+ A C) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0)))))))
          t_2))
        (t_4
         (/
          (* (sqrt (* 2.0 (* F t_0))) (- (sqrt (+ C (+ A (hypot B (- A C)))))))
          t_0)))
   (if (<= t_3 -1e-204)
     t_4
     (if (<= t_3 2e+17)
       (/
        (- (sqrt (* 2.0 (* (* F t_1) (fma 2.0 C (* -0.5 (/ (* B B) A)))))))
        t_1)
       (if (<= t_3 INFINITY)
         t_4
         (* (/ (sqrt 2.0) B) (* (sqrt F) (- (sqrt (+ A (hypot B A)))))))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double t_1 = fma(C, (A * -4.0), (B * B));
	double t_2 = pow(B, 2.0) - ((4.0 * A) * C);
	double t_3 = -sqrt(((2.0 * (t_2 * F)) * ((A + C) + sqrt((pow(B, 2.0) + pow((A - C), 2.0)))))) / t_2;
	double t_4 = (sqrt((2.0 * (F * t_0))) * -sqrt((C + (A + hypot(B, (A - C)))))) / t_0;
	double tmp;
	if (t_3 <= -1e-204) {
		tmp = t_4;
	} else if (t_3 <= 2e+17) {
		tmp = -sqrt((2.0 * ((F * t_1) * fma(2.0, C, (-0.5 * ((B * B) / A)))))) / t_1;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_4;
	} else {
		tmp = (sqrt(2.0) / B) * (sqrt(F) * -sqrt((A + hypot(B, A))));
	}
	return tmp;
}

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	t_1 = fma(C, Float64(A * -4.0), Float64(B * B))
	t_2 = Float64((B ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_3 = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_2 * F)) * Float64(Float64(A + C) + sqrt(Float64((B ^ 2.0) + (Float64(A - C) ^ 2.0))))))) / t_2)
	t_4 = Float64(Float64(sqrt(Float64(2.0 * Float64(F * t_0))) * Float64(-sqrt(Float64(C + Float64(A + hypot(B, Float64(A - C))))))) / t_0)
	tmp = 0.0
	if (t_3 <= -1e-204)
		tmp = t_4;
	elseif (t_3 <= 2e+17)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(F * t_1) * fma(2.0, C, Float64(-0.5 * Float64(Float64(B * B) / A))))))) / t_1);
	elseif (t_3 <= Inf)
		tmp = t_4;
	else
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(sqrt(F) * Float64(-sqrt(Float64(A + hypot(B, A))))));
	end
	return tmp
end

Wolfram

NOTE: B should be positive 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[(C * N[(A * -4.0), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$2 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[Sqrt[N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(C + N[(A + N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[t$95$3, -1e-204], t$95$4, If[LessEqual[t$95$3, 2e+17], N[((-N[Sqrt[N[(2.0 * N[(N[(F * t$95$1), $MachinePrecision] * N[(2.0 * C + N[(-0.5 * N[(N[(B * B), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$4, N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[N[(A + N[Sqrt[B ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < -1e-204 or 2e17 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < +inf.0

   1. Initial program 40.8%

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

      1. associate-*l* 40.8%

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

      2. unpow2 40.8%

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

      3. +-commutative 40.8%

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

      4. unpow2 40.8%

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

      5. associate-*l* 40.8%

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

      6. unpow2 40.8%

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

   3. Simplified 40.8%

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

      1. sqrt-prod 44.2%

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

      2. *-commutative 44.2%

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

      3. *-commutative 44.2%

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

      4. associate-+l+ 44.2%

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

      5. unpow2 44.2%

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

      6. hypot-udef 65.1%

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

      7. associate-+r+ 65.0%

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

      8. +-commutative 65.0%

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

      9. associate-+r+ 65.8%

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

   5. Applied egg-rr 65.8%

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

   if -1e-204 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < 2e17

   1. Initial program 6.6%

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

   3. Simplified 12.4%

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

   4. Taylor expanded in A around -inf 40.1%

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

      1. fma-def 40.1%

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

      2. unpow2 40.1%

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

   6. Simplified 40.1%

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

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

   1. Initial program 0.0%

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

   3. Simplified 0.6%

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

   4. Taylor expanded in C around 0 1.8%

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

      1. mul-1-neg 1.8%

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

      2. distribute-rgt-neg-in 1.8%

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

      3. unpow2 1.8%

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

      4. unpow2 1.8%

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

      5. hypot-def 24.5%

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

   6. Simplified 24.5%

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

      1. sqrt-prod 34.2%

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

   8. Applied egg-rr 34.2%

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

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq -1 \cdot 10^{-204}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \left(-\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq 2 \cdot 10^{+17}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\right) \cdot \mathsf{fma}\left(2, C, -0.5 \cdot \frac{B \cdot B}{A}\right)\right)}}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \left(-\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{A + \mathsf{hypot}\left(B, A\right)}\right)\right)\\ \end{array} \]

Alternative 2: 53.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq 1.18 \cdot 10^{+100}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot t_0\right)} \cdot \left(-\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{A + \mathsf{hypot}\left(B, A\right)}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* A C)))))
   (if (<= B 1.18e+100)
     (/
      (* (sqrt (* 2.0 (* F t_0))) (- (sqrt (+ C (+ A (hypot B (- A C)))))))
      t_0)
     (* (/ (sqrt 2.0) B) (* (sqrt F) (- (sqrt (+ A (hypot B A)))))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (B <= 1.18e+100) {
		tmp = (sqrt((2.0 * (F * t_0))) * -sqrt((C + (A + hypot(B, (A - C)))))) / t_0;
	} else {
		tmp = (sqrt(2.0) / B) * (sqrt(F) * -sqrt((A + hypot(B, A))));
	}
	return tmp;
}

Java

B = Math.abs(B);
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 <= 1.18e+100) {
		tmp = (Math.sqrt((2.0 * (F * t_0))) * -Math.sqrt((C + (A + Math.hypot(B, (A - C)))))) / t_0;
	} else {
		tmp = (Math.sqrt(2.0) / B) * (Math.sqrt(F) * -Math.sqrt((A + Math.hypot(B, A))));
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (A * C))
	tmp = 0
	if B <= 1.18e+100:
		tmp = (math.sqrt((2.0 * (F * t_0))) * -math.sqrt((C + (A + math.hypot(B, (A - C)))))) / t_0
	else:
		tmp = (math.sqrt(2.0) / B) * (math.sqrt(F) * -math.sqrt((A + math.hypot(B, A))))
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	tmp = 0.0
	if (B <= 1.18e+100)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(F * t_0))) * Float64(-sqrt(Float64(C + Float64(A + hypot(B, Float64(A - C))))))) / t_0);
	else
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(sqrt(F) * Float64(-sqrt(Float64(A + hypot(B, A))))));
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (A * C));
	tmp = 0.0;
	if (B <= 1.18e+100)
		tmp = (sqrt((2.0 * (F * t_0))) * -sqrt((C + (A + hypot(B, (A - C)))))) / t_0;
	else
		tmp = (sqrt(2.0) / B) * (sqrt(F) * -sqrt((A + hypot(B, A))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive 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, 1.18e+100], N[(N[(N[Sqrt[N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(C + N[(A + N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[N[(A + N[Sqrt[B ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 1.18e100

   1. Initial program 23.8%

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

      1. associate-*l* 23.8%

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

      2. unpow2 23.8%

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

      3. +-commutative 23.8%

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

      4. unpow2 23.8%

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

      5. associate-*l* 23.8%

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

      6. unpow2 23.8%

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

   3. Simplified 23.8%

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

      1. sqrt-prod 25.7%

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

      2. *-commutative 25.7%

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

      3. *-commutative 25.7%

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

      4. associate-+l+ 26.1%

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

      5. unpow2 26.1%

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

      6. hypot-udef 37.9%

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

      7. associate-+r+ 37.4%

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

      8. +-commutative 37.4%

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

      9. associate-+r+ 38.5%

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

   5. Applied egg-rr 38.5%

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

   if 1.18e100 < B

   1. Initial program 2.9%

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

   3. Simplified 2.9%

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

   4. Taylor expanded in C around 0 7.2%

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

      1. mul-1-neg 7.2%

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

      2. distribute-rgt-neg-in 7.2%

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

      3. unpow2 7.2%

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

      4. unpow2 7.2%

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

      5. hypot-def 55.2%

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

   6. Simplified 55.2%

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

      1. sqrt-prod 81.2%

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

   8. Applied egg-rr 81.2%

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

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.18 \cdot 10^{+100}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \left(-\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{A + \mathsf{hypot}\left(B, A\right)}\right)\right)\\ \end{array} \]

Alternative 3: 44.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq 9.4 \cdot 10^{+100}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot t_0\right)} \cdot \left(-\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \left(-\sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)}\right)}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* A C)))))
   (if (<= B 9.4e+100)
     (/
      (* (sqrt (* 2.0 (* F t_0))) (- (sqrt (+ C (+ A (hypot B (- A C)))))))
      t_0)
     (/ (* (sqrt 2.0) (- (sqrt (* F (+ A (hypot B A)))))) B))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (B <= 9.4e+100) {
		tmp = (sqrt((2.0 * (F * t_0))) * -sqrt((C + (A + hypot(B, (A - C)))))) / t_0;
	} else {
		tmp = (sqrt(2.0) * -sqrt((F * (A + hypot(B, A))))) / B;
	}
	return tmp;
}

Java

B = Math.abs(B);
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 <= 9.4e+100) {
		tmp = (Math.sqrt((2.0 * (F * t_0))) * -Math.sqrt((C + (A + Math.hypot(B, (A - C)))))) / t_0;
	} else {
		tmp = (Math.sqrt(2.0) * -Math.sqrt((F * (A + Math.hypot(B, A))))) / B;
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (A * C))
	tmp = 0
	if B <= 9.4e+100:
		tmp = (math.sqrt((2.0 * (F * t_0))) * -math.sqrt((C + (A + math.hypot(B, (A - C)))))) / t_0
	else:
		tmp = (math.sqrt(2.0) * -math.sqrt((F * (A + math.hypot(B, A))))) / B
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	tmp = 0.0
	if (B <= 9.4e+100)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(F * t_0))) * Float64(-sqrt(Float64(C + Float64(A + hypot(B, Float64(A - C))))))) / t_0);
	else
		tmp = Float64(Float64(sqrt(2.0) * Float64(-sqrt(Float64(F * Float64(A + hypot(B, A)))))) / B);
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (A * C));
	tmp = 0.0;
	if (B <= 9.4e+100)
		tmp = (sqrt((2.0 * (F * t_0))) * -sqrt((C + (A + hypot(B, (A - C)))))) / t_0;
	else
		tmp = (sqrt(2.0) * -sqrt((F * (A + hypot(B, A))))) / B;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive 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, 9.4e+100], N[(N[(N[Sqrt[N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(C + N[(A + N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F * N[(A + N[Sqrt[B ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 9.4e100

   1. Initial program 23.8%

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

      1. associate-*l* 23.8%

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

      2. unpow2 23.8%

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

      3. +-commutative 23.8%

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

      4. unpow2 23.8%

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

      5. associate-*l* 23.8%

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

      6. unpow2 23.8%

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

   3. Simplified 23.8%

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

      1. sqrt-prod 25.7%

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

      2. *-commutative 25.7%

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

      3. *-commutative 25.7%

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

      4. associate-+l+ 26.1%

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

      5. unpow2 26.1%

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

      6. hypot-udef 37.9%

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

      7. associate-+r+ 37.4%

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

      8. +-commutative 37.4%

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

      9. associate-+r+ 38.5%

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

   5. Applied egg-rr 38.5%

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

   if 9.4e100 < B

   1. Initial program 2.9%

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

   3. Simplified 2.9%

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

   4. Taylor expanded in C around 0 7.2%

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

      1. mul-1-neg 7.2%

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

      2. distribute-rgt-neg-in 7.2%

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

      3. unpow2 7.2%

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

      4. unpow2 7.2%

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

      5. hypot-def 55.2%

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

   6. Simplified 55.2%

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

      1. associate-*l/ 55.3%

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

   8. Applied egg-rr 55.3%

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

4. Final simplification 41.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 9.4 \cdot 10^{+100}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \left(-\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \left(-\sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)}\right)}{B}\\ \end{array} \]

Alternative 4: 37.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ t_1 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_2 := \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\\ \mathbf{if}\;B \leq 1.035 \cdot 10^{-283}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot t_1\right)\right) \cdot \left(A + A\right)}}{t_1}\\ \mathbf{elif}\;B \leq 1.08 \cdot 10^{-74}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_2\right) \cdot \left(C + C\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 2.3 \cdot 10^{+46}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \left(-\sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)}\right)}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* A C))))
        (t_1 (fma B B (* C (* A -4.0))))
        (t_2 (fma C (* A -4.0) (* B B))))
   (if (<= B 1.035e-283)
     (/ (- (sqrt (* (* 2.0 (* F t_1)) (+ A A)))) t_1)
     (if (<= B 1.08e-74)
       (/ (- (sqrt (* 2.0 (* (* F t_2) (+ C C))))) t_2)
       (if (<= B 2.3e+46)
         (/ (- (sqrt (* 2.0 (* (* F t_0) (+ C (+ A (hypot B (- A C)))))))) t_0)
         (/ (* (sqrt 2.0) (- (sqrt (* F (+ A (hypot B A)))))) B))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double t_1 = fma(B, B, (C * (A * -4.0)));
	double t_2 = fma(C, (A * -4.0), (B * B));
	double tmp;
	if (B <= 1.035e-283) {
		tmp = -sqrt(((2.0 * (F * t_1)) * (A + A))) / t_1;
	} else if (B <= 1.08e-74) {
		tmp = -sqrt((2.0 * ((F * t_2) * (C + C)))) / t_2;
	} else if (B <= 2.3e+46) {
		tmp = -sqrt((2.0 * ((F * t_0) * (C + (A + hypot(B, (A - C))))))) / t_0;
	} else {
		tmp = (sqrt(2.0) * -sqrt((F * (A + hypot(B, A))))) / B;
	}
	return tmp;
}

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	t_1 = fma(B, B, Float64(C * Float64(A * -4.0)))
	t_2 = fma(C, Float64(A * -4.0), Float64(B * B))
	tmp = 0.0
	if (B <= 1.035e-283)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(F * t_1)) * Float64(A + A)))) / t_1);
	elseif (B <= 1.08e-74)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(F * t_2) * Float64(C + C))))) / t_2);
	elseif (B <= 2.3e+46)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(F * t_0) * Float64(C + Float64(A + hypot(B, Float64(A - C)))))))) / t_0);
	else
		tmp = Float64(Float64(sqrt(2.0) * Float64(-sqrt(Float64(F * Float64(A + hypot(B, A)))))) / B);
	end
	return tmp
end

Wolfram

NOTE: B should be positive 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[(B * B + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(C * N[(A * -4.0), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, 1.035e-283], N[((-N[Sqrt[N[(N[(2.0 * N[(F * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(A + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], If[LessEqual[B, 1.08e-74], N[((-N[Sqrt[N[(2.0 * N[(N[(F * t$95$2), $MachinePrecision] * N[(C + C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision], If[LessEqual[B, 2.3e+46], N[((-N[Sqrt[N[(2.0 * N[(N[(F * t$95$0), $MachinePrecision] * N[(C + N[(A + N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F * N[(A + N[Sqrt[B ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / B), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 1.035e-283

   1. Initial program 14.0%

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

   3. Simplified 21.6%

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

   4. Taylor expanded in A around inf 13.1%

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

      1. distribute-rgt1-in 13.1%

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

      2. metadata-eval 13.1%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot F\right)\right) \cdot \left(A + \left(A + \color{blue}{0} \cdot C\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]

      3. mul0-lft 13.1%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot F\right)\right) \cdot \left(A + \left(A + \color{blue}{0}\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]

   6. Simplified 13.1%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\left(A + 0\right)}\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]

   if 1.035e-283 < B < 1.0799999999999999e-74

   1. Initial program 25.5%

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

   3. Simplified 33.6%

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

   4. Taylor expanded in A around -inf 40.9%

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

   if 1.0799999999999999e-74 < B < 2.3000000000000001e46

   1. Initial program 48.3%

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

      1. associate-*l* 48.3%

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

      2. unpow2 48.3%

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

      3. +-commutative 48.3%

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

      4. unpow2 48.3%

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

      5. associate-*l* 48.3%

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

      6. unpow2 48.3%

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

   3. Simplified 48.3%

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

      1. distribute-frac-neg 48.3%

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

   5. Applied egg-rr 57.4%

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

   if 2.3000000000000001e46 < B

   1. Initial program 10.5%

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

   3. Simplified 12.5%

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

   4. Taylor expanded in C around 0 16.1%

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

      1. mul-1-neg 16.1%

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

      2. distribute-rgt-neg-in 16.1%

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

      3. unpow2 16.1%

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

      4. unpow2 16.1%

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

      5. hypot-def 55.7%

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

   6. Simplified 55.7%

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

      1. associate-*l/ 55.7%

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

   8. Applied egg-rr 55.7%

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

4. Final simplification 33.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.035 \cdot 10^{-283}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)\right) \cdot \left(A + A\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 1.08 \cdot 10^{-74}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\right) \cdot \left(C + C\right)\right)}}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 2.3 \cdot 10^{+46}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \left(-\sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)}\right)}{B}\\ \end{array} \]

Alternative 5: 38.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\\ t_1 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq 1.35 \cdot 10^{-74}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(C + C\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 1.55 \cdot 10^{+47}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_1\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \left(-\sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)}\right)}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma C (* A -4.0) (* B B))) (t_1 (- (* B B) (* 4.0 (* A C)))))
   (if (<= B 1.35e-74)
     (/ (- (sqrt (* 2.0 (* (* F t_0) (+ C C))))) t_0)
     (if (<= B 1.55e+47)
       (/ (- (sqrt (* 2.0 (* (* F t_1) (+ C (+ A (hypot B (- A C)))))))) t_1)
       (/ (* (sqrt 2.0) (- (sqrt (* F (+ A (hypot B A)))))) B)))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = fma(C, (A * -4.0), (B * B));
	double t_1 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (B <= 1.35e-74) {
		tmp = -sqrt((2.0 * ((F * t_0) * (C + C)))) / t_0;
	} else if (B <= 1.55e+47) {
		tmp = -sqrt((2.0 * ((F * t_1) * (C + (A + hypot(B, (A - C))))))) / t_1;
	} else {
		tmp = (sqrt(2.0) * -sqrt((F * (A + hypot(B, A))))) / B;
	}
	return tmp;
}

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = fma(C, Float64(A * -4.0), Float64(B * B))
	t_1 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	tmp = 0.0
	if (B <= 1.35e-74)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(F * t_0) * Float64(C + C))))) / t_0);
	elseif (B <= 1.55e+47)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(F * t_1) * Float64(C + Float64(A + hypot(B, Float64(A - C)))))))) / t_1);
	else
		tmp = Float64(Float64(sqrt(2.0) * Float64(-sqrt(Float64(F * Float64(A + hypot(B, A)))))) / B);
	end
	return tmp
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(C * N[(A * -4.0), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, 1.35e-74], N[((-N[Sqrt[N[(2.0 * N[(N[(F * t$95$0), $MachinePrecision] * N[(C + C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[B, 1.55e+47], N[((-N[Sqrt[N[(2.0 * N[(N[(F * t$95$1), $MachinePrecision] * N[(C + N[(A + N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F * N[(A + N[Sqrt[B ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / B), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.35000000000000009e-74

   1. Initial program 17.5%

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

   3. Simplified 25.1%

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

   4. Taylor expanded in A around -inf 22.5%

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

   if 1.35000000000000009e-74 < B < 1.55e47

   1. Initial program 48.3%

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

      1. associate-*l* 48.3%

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

      2. unpow2 48.3%

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

      3. +-commutative 48.3%

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

      4. unpow2 48.3%

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

      5. associate-*l* 48.3%

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

      6. unpow2 48.3%

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

   3. Simplified 48.3%

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

      1. distribute-frac-neg 48.3%

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

   5. Applied egg-rr 57.4%

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

   if 1.55e47 < B

   1. Initial program 10.5%

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

   3. Simplified 12.5%

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

   4. Taylor expanded in C around 0 16.1%

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

      1. mul-1-neg 16.1%

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

      2. distribute-rgt-neg-in 16.1%

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

      3. unpow2 16.1%

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

      4. unpow2 16.1%

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

      5. hypot-def 55.7%

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

   6. Simplified 55.7%

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

      1. associate-*l/ 55.7%

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

   8. Applied egg-rr 55.7%

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

4. Final simplification 33.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.35 \cdot 10^{-74}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\right) \cdot \left(C + C\right)\right)}}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 1.55 \cdot 10^{+47}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \left(-\sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)}\right)}{B}\\ \end{array} \]

Alternative 6: 39.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq 8.2 \cdot 10^{+49}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* A C)))))
   (if (<= B 8.2e+49)
     (/ (- (sqrt (* 2.0 (* (* F t_0) (+ C (+ A (hypot B (- A C)))))))) t_0)
     (* (/ (sqrt 2.0) B) (- (sqrt (* F (+ A (hypot B A)))))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (B <= 8.2e+49) {
		tmp = -sqrt((2.0 * ((F * t_0) * (C + (A + hypot(B, (A - C))))))) / t_0;
	} else {
		tmp = (sqrt(2.0) / B) * -sqrt((F * (A + hypot(B, A))));
	}
	return tmp;
}

Java

B = Math.abs(B);
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 <= 8.2e+49) {
		tmp = -Math.sqrt((2.0 * ((F * t_0) * (C + (A + Math.hypot(B, (A - C))))))) / t_0;
	} else {
		tmp = (Math.sqrt(2.0) / B) * -Math.sqrt((F * (A + Math.hypot(B, A))));
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (A * C))
	tmp = 0
	if B <= 8.2e+49:
		tmp = -math.sqrt((2.0 * ((F * t_0) * (C + (A + math.hypot(B, (A - C))))))) / t_0
	else:
		tmp = (math.sqrt(2.0) / B) * -math.sqrt((F * (A + math.hypot(B, A))))
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	tmp = 0.0
	if (B <= 8.2e+49)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(F * t_0) * Float64(C + Float64(A + hypot(B, Float64(A - C)))))))) / t_0);
	else
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(-sqrt(Float64(F * Float64(A + hypot(B, A))))));
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (A * C));
	tmp = 0.0;
	if (B <= 8.2e+49)
		tmp = -sqrt((2.0 * ((F * t_0) * (C + (A + hypot(B, (A - C))))))) / t_0;
	else
		tmp = (sqrt(2.0) / B) * -sqrt((F * (A + hypot(B, A))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive 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, 8.2e+49], N[((-N[Sqrt[N[(2.0 * N[(N[(F * t$95$0), $MachinePrecision] * N[(C + N[(A + N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * (-N[Sqrt[N[(F * N[(A + N[Sqrt[B ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 8.2e49

   1. Initial program 22.8%

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

      1. associate-*l* 22.8%

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

      2. unpow2 22.8%

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

      3. +-commutative 22.8%

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

      4. unpow2 22.8%

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

      5. associate-*l* 22.8%

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

      6. unpow2 22.8%

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

   3. Simplified 22.8%

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

      1. distribute-frac-neg 22.8%

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

   5. Applied egg-rr 30.8%

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

   if 8.2e49 < B

   1. Initial program 10.5%

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

   3. Simplified 12.5%

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

   4. Taylor expanded in C around 0 16.1%

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

      1. mul-1-neg 16.1%

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

      2. distribute-rgt-neg-in 16.1%

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

      3. unpow2 16.1%

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

      4. unpow2 16.1%

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

      5. hypot-def 55.7%

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

   6. Simplified 55.7%

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

4. Final simplification 35.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 8.2 \cdot 10^{+49}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)}\right)\\ \end{array} \]

Alternative 7: 39.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq 2.5 \cdot 10^{+48}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \left(-\sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)}\right)}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* A C)))))
   (if (<= B 2.5e+48)
     (/ (- (sqrt (* 2.0 (* (* F t_0) (+ C (+ A (hypot B (- A C)))))))) t_0)
     (/ (* (sqrt 2.0) (- (sqrt (* F (+ A (hypot B A)))))) B))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (B <= 2.5e+48) {
		tmp = -sqrt((2.0 * ((F * t_0) * (C + (A + hypot(B, (A - C))))))) / t_0;
	} else {
		tmp = (sqrt(2.0) * -sqrt((F * (A + hypot(B, A))))) / B;
	}
	return tmp;
}

Java

B = Math.abs(B);
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 <= 2.5e+48) {
		tmp = -Math.sqrt((2.0 * ((F * t_0) * (C + (A + Math.hypot(B, (A - C))))))) / t_0;
	} else {
		tmp = (Math.sqrt(2.0) * -Math.sqrt((F * (A + Math.hypot(B, A))))) / B;
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (A * C))
	tmp = 0
	if B <= 2.5e+48:
		tmp = -math.sqrt((2.0 * ((F * t_0) * (C + (A + math.hypot(B, (A - C))))))) / t_0
	else:
		tmp = (math.sqrt(2.0) * -math.sqrt((F * (A + math.hypot(B, A))))) / B
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	tmp = 0.0
	if (B <= 2.5e+48)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(F * t_0) * Float64(C + Float64(A + hypot(B, Float64(A - C)))))))) / t_0);
	else
		tmp = Float64(Float64(sqrt(2.0) * Float64(-sqrt(Float64(F * Float64(A + hypot(B, A)))))) / B);
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (A * C));
	tmp = 0.0;
	if (B <= 2.5e+48)
		tmp = -sqrt((2.0 * ((F * t_0) * (C + (A + hypot(B, (A - C))))))) / t_0;
	else
		tmp = (sqrt(2.0) * -sqrt((F * (A + hypot(B, A))))) / B;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive 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, 2.5e+48], N[((-N[Sqrt[N[(2.0 * N[(N[(F * t$95$0), $MachinePrecision] * N[(C + N[(A + N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F * N[(A + N[Sqrt[B ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 2.49999999999999987e48

   1. Initial program 22.8%

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

      1. associate-*l* 22.8%

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

      2. unpow2 22.8%

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

      3. +-commutative 22.8%

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

      4. unpow2 22.8%

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

      5. associate-*l* 22.8%

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

      6. unpow2 22.8%

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

   3. Simplified 22.8%

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

      1. distribute-frac-neg 22.8%

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

   5. Applied egg-rr 30.8%

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

   if 2.49999999999999987e48 < B

   1. Initial program 10.5%

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

   3. Simplified 12.5%

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

   4. Taylor expanded in C around 0 16.1%

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

      1. mul-1-neg 16.1%

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

      2. distribute-rgt-neg-in 16.1%

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

      3. unpow2 16.1%

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

      4. unpow2 16.1%

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

      5. hypot-def 55.7%

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

   6. Simplified 55.7%

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

      1. associate-*l/ 55.7%

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

   8. Applied egg-rr 55.7%

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

4. Final simplification 35.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.5 \cdot 10^{+48}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \left(-\sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)}\right)}{B}\\ \end{array} \]

Alternative 8: 31.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ t_1 := 2 \cdot \left(F \cdot t_0\right)\\ \mathbf{if}\;B \leq 1.25 \cdot 10^{-128}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(\left(A + C\right) + \left(C + \left(0.5 \cdot \frac{B \cdot B}{C} - A\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 2.2 \cdot 10^{+47}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(B + A\right)}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* A C)))) (t_1 (* 2.0 (* F t_0))))
   (if (<= B 1.25e-128)
     (/ (- (sqrt (* t_1 (+ (+ A C) (+ C (- (* 0.5 (/ (* B B) C)) A)))))) t_0)
     (if (<= B 2.2e+47)
       (/ (- (sqrt (* t_1 (+ (+ A C) (hypot B A))))) t_0)
       (* (/ (sqrt 2.0) B) (- (sqrt (* F (+ B A)))))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double t_1 = 2.0 * (F * t_0);
	double tmp;
	if (B <= 1.25e-128) {
		tmp = -sqrt((t_1 * ((A + C) + (C + ((0.5 * ((B * B) / C)) - A))))) / t_0;
	} else if (B <= 2.2e+47) {
		tmp = -sqrt((t_1 * ((A + C) + hypot(B, A)))) / t_0;
	} else {
		tmp = (sqrt(2.0) / B) * -sqrt((F * (B + A)));
	}
	return tmp;
}

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double t_1 = 2.0 * (F * t_0);
	double tmp;
	if (B <= 1.25e-128) {
		tmp = -Math.sqrt((t_1 * ((A + C) + (C + ((0.5 * ((B * B) / C)) - A))))) / t_0;
	} else if (B <= 2.2e+47) {
		tmp = -Math.sqrt((t_1 * ((A + C) + Math.hypot(B, A)))) / t_0;
	} else {
		tmp = (Math.sqrt(2.0) / B) * -Math.sqrt((F * (B + A)));
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (A * C))
	t_1 = 2.0 * (F * t_0)
	tmp = 0
	if B <= 1.25e-128:
		tmp = -math.sqrt((t_1 * ((A + C) + (C + ((0.5 * ((B * B) / C)) - A))))) / t_0
	elif B <= 2.2e+47:
		tmp = -math.sqrt((t_1 * ((A + C) + math.hypot(B, A)))) / t_0
	else:
		tmp = (math.sqrt(2.0) / B) * -math.sqrt((F * (B + A)))
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	t_1 = Float64(2.0 * Float64(F * t_0))
	tmp = 0.0
	if (B <= 1.25e-128)
		tmp = Float64(Float64(-sqrt(Float64(t_1 * Float64(Float64(A + C) + Float64(C + Float64(Float64(0.5 * Float64(Float64(B * B) / C)) - A)))))) / t_0);
	elseif (B <= 2.2e+47)
		tmp = Float64(Float64(-sqrt(Float64(t_1 * Float64(Float64(A + C) + hypot(B, A))))) / t_0);
	else
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(-sqrt(Float64(F * Float64(B + A)))));
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (A * C));
	t_1 = 2.0 * (F * t_0);
	tmp = 0.0;
	if (B <= 1.25e-128)
		tmp = -sqrt((t_1 * ((A + C) + (C + ((0.5 * ((B * B) / C)) - A))))) / t_0;
	elseif (B <= 2.2e+47)
		tmp = -sqrt((t_1 * ((A + C) + hypot(B, A)))) / t_0;
	else
		tmp = (sqrt(2.0) / B) * -sqrt((F * (B + A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive 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[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, 1.25e-128], N[((-N[Sqrt[N[(t$95$1 * N[(N[(A + C), $MachinePrecision] + N[(C + N[(N[(0.5 * N[(N[(B * B), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision] - A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[B, 2.2e+47], N[((-N[Sqrt[N[(t$95$1 * N[(N[(A + C), $MachinePrecision] + N[Sqrt[B ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * (-N[Sqrt[N[(F * N[(B + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.25e-128

   1. Initial program 15.9%

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

      1. associate-*l* 15.9%

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

      2. unpow2 15.9%

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

      3. +-commutative 15.9%

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

      4. unpow2 15.9%

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

      5. associate-*l* 15.9%

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

      6. unpow2 15.9%

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

   3. Simplified 15.9%

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

   4. Taylor expanded in C around inf 11.4%

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

      1. mul-1-neg 11.4%

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

      2. unsub-neg 11.4%

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

   6. Simplified 12.0%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \color{blue}{\left(C + \left(0.5 \cdot \frac{B \cdot B + 0}{C} - A\right)\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   if 1.25e-128 < B < 2.1999999999999999e47

   1. Initial program 42.8%

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

      1. associate-*l* 42.8%

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

      2. unpow2 42.8%

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

      3. +-commutative 42.8%

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

      4. unpow2 42.8%

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

      5. associate-*l* 42.8%

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

      6. unpow2 42.8%

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

   3. Simplified 42.8%

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

   4. Taylor expanded in C around 0 36.2%

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

      1. unpow2 36.2%

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

      2. unpow2 36.2%

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

      3. hypot-def 41.5%

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

   6. Simplified 41.5%

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

   if 2.1999999999999999e47 < B

   1. Initial program 10.5%

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

   3. Simplified 12.5%

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

   4. Taylor expanded in C around 0 16.1%

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

      1. mul-1-neg 16.1%

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

      2. distribute-rgt-neg-in 16.1%

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

      3. unpow2 16.1%

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

      4. unpow2 16.1%

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

      5. hypot-def 55.7%

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

   6. Simplified 55.7%

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

   7. Taylor expanded in A around 0 53.7%

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

      1. +-commutative 53.7%

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

   9. Simplified 53.7%

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

4. Final simplification 26.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.25 \cdot 10^{-128}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right)\right) \cdot \left(\left(A + C\right) + \left(C + \left(0.5 \cdot \frac{B \cdot B}{C} - A\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 2.2 \cdot 10^{+47}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(B + A\right)}\right)\\ \end{array} \]

Alternative 9: 38.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq 5.2 \cdot 10^{+84}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(B + A\right)}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* A C)))))
   (if (<= B 5.2e+84)
     (/ (- (sqrt (* 2.0 (* (* F t_0) (+ C (+ A (hypot B (- A C)))))))) t_0)
     (* (/ (sqrt 2.0) B) (- (sqrt (* F (+ B A))))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (B <= 5.2e+84) {
		tmp = -sqrt((2.0 * ((F * t_0) * (C + (A + hypot(B, (A - C))))))) / t_0;
	} else {
		tmp = (sqrt(2.0) / B) * -sqrt((F * (B + A)));
	}
	return tmp;
}

Java

B = Math.abs(B);
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 <= 5.2e+84) {
		tmp = -Math.sqrt((2.0 * ((F * t_0) * (C + (A + Math.hypot(B, (A - C))))))) / t_0;
	} else {
		tmp = (Math.sqrt(2.0) / B) * -Math.sqrt((F * (B + A)));
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (A * C))
	tmp = 0
	if B <= 5.2e+84:
		tmp = -math.sqrt((2.0 * ((F * t_0) * (C + (A + math.hypot(B, (A - C))))))) / t_0
	else:
		tmp = (math.sqrt(2.0) / B) * -math.sqrt((F * (B + A)))
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	tmp = 0.0
	if (B <= 5.2e+84)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(F * t_0) * Float64(C + Float64(A + hypot(B, Float64(A - C)))))))) / t_0);
	else
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(-sqrt(Float64(F * Float64(B + A)))));
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (A * C));
	tmp = 0.0;
	if (B <= 5.2e+84)
		tmp = -sqrt((2.0 * ((F * t_0) * (C + (A + hypot(B, (A - C))))))) / t_0;
	else
		tmp = (sqrt(2.0) / B) * -sqrt((F * (B + A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive 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, 5.2e+84], N[((-N[Sqrt[N[(2.0 * N[(N[(F * t$95$0), $MachinePrecision] * N[(C + N[(A + N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * (-N[Sqrt[N[(F * N[(B + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 5.2000000000000002e84

   1. Initial program 24.0%

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

      1. associate-*l* 24.0%

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

      2. unpow2 24.0%

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

      3. +-commutative 24.0%

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

      4. unpow2 24.0%

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

      5. associate-*l* 24.0%

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

      6. unpow2 24.0%

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

   3. Simplified 24.0%

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

      1. distribute-frac-neg 24.0%

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

   5. Applied egg-rr 32.1%

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

   if 5.2000000000000002e84 < B

   1. Initial program 2.9%

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

   3. Simplified 2.9%

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

   4. Taylor expanded in C around 0 9.3%

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

      1. mul-1-neg 9.3%

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

      2. distribute-rgt-neg-in 9.3%

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

      3. unpow2 9.3%

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

      4. unpow2 9.3%

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

      5. hypot-def 55.1%

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

   6. Simplified 55.1%

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

   7. Taylor expanded in A around 0 54.9%

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

      1. +-commutative 54.9%

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

   9. Simplified 54.9%

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

4. Final simplification 35.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 5.2 \cdot 10^{+84}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(B + A\right)}\right)\\ \end{array} \]

Alternative 10: 37.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;F \leq -5.2 \cdot 10^{-296}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot t_0\right)\right) \cdot \left(A + \left(A + C\right)\right)}}{t_0}\\ \mathbf{elif}\;F \leq 1.55 \cdot 10^{+24}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{B \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{\frac{F}{B}}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* A C)))))
   (if (<= F -5.2e-296)
     (/ (- (sqrt (* (* 2.0 (* F t_0)) (+ A (+ A C))))) t_0)
     (if (<= F 1.55e+24)
       (* (/ (sqrt 2.0) B) (- (sqrt (* B F))))
       (- (* (sqrt 2.0) (sqrt (/ F B))))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (F <= -5.2e-296) {
		tmp = -sqrt(((2.0 * (F * t_0)) * (A + (A + C)))) / t_0;
	} else if (F <= 1.55e+24) {
		tmp = (sqrt(2.0) / B) * -sqrt((B * F));
	} else {
		tmp = -(sqrt(2.0) * sqrt((F / B)));
	}
	return tmp;
}

Fortran

NOTE: B should be positive 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 (f <= (-5.2d-296)) then
        tmp = -sqrt(((2.0d0 * (f * t_0)) * (a + (a + c)))) / t_0
    else if (f <= 1.55d+24) then
        tmp = (sqrt(2.0d0) / b) * -sqrt((b * f))
    else
        tmp = -(sqrt(2.0d0) * sqrt((f / b)))
    end if
    code = tmp
end function

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (F <= -5.2e-296) {
		tmp = -Math.sqrt(((2.0 * (F * t_0)) * (A + (A + C)))) / t_0;
	} else if (F <= 1.55e+24) {
		tmp = (Math.sqrt(2.0) / B) * -Math.sqrt((B * F));
	} else {
		tmp = -(Math.sqrt(2.0) * Math.sqrt((F / B)));
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (A * C))
	tmp = 0
	if F <= -5.2e-296:
		tmp = -math.sqrt(((2.0 * (F * t_0)) * (A + (A + C)))) / t_0
	elif F <= 1.55e+24:
		tmp = (math.sqrt(2.0) / B) * -math.sqrt((B * F))
	else:
		tmp = -(math.sqrt(2.0) * math.sqrt((F / B)))
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	tmp = 0.0
	if (F <= -5.2e-296)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(F * t_0)) * Float64(A + Float64(A + C))))) / t_0);
	elseif (F <= 1.55e+24)
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(-sqrt(Float64(B * F))));
	else
		tmp = Float64(-Float64(sqrt(2.0) * sqrt(Float64(F / B))));
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (A * C));
	tmp = 0.0;
	if (F <= -5.2e-296)
		tmp = -sqrt(((2.0 * (F * t_0)) * (A + (A + C)))) / t_0;
	elseif (F <= 1.55e+24)
		tmp = (sqrt(2.0) / B) * -sqrt((B * F));
	else
		tmp = -(sqrt(2.0) * sqrt((F / B)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive 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[F, -5.2e-296], N[((-N[Sqrt[N[(N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(A + N[(A + C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[F, 1.55e+24], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * (-N[Sqrt[N[(B * F), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], (-N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.2000000000000001e-296

   1. Initial program 24.8%

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

      1. associate-*l* 24.8%

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

      2. unpow2 24.8%

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

      3. +-commutative 24.8%

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

      4. unpow2 24.8%

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

      5. associate-*l* 24.8%

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

      6. unpow2 24.8%

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

   3. Simplified 24.8%

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

   4. Taylor expanded in A around inf 25.4%

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

   if -5.2000000000000001e-296 < F < 1.55000000000000005e24

   1. Initial program 22.5%

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

   3. Simplified 30.8%

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

   4. Taylor expanded in C around 0 13.3%

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

      1. mul-1-neg 13.3%

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

      2. distribute-rgt-neg-in 13.3%

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

      3. unpow2 13.3%

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

      4. unpow2 13.3%

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

      5. hypot-def 29.1%

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

   6. Simplified 29.1%

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

   7. Taylor expanded in A around 0 24.6%

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

   if 1.55000000000000005e24 < F

   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 18.1%

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

   4. Taylor expanded in C around 0 11.4%

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

      1. mul-1-neg 11.4%

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

      2. distribute-rgt-neg-in 11.4%

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

      3. unpow2 11.4%

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

      4. unpow2 11.4%

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

      5. hypot-def 14.8%

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

   6. Simplified 14.8%

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

   7. Taylor expanded in A around 0 21.5%

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

      1. associate-*r* 21.5%

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

      2. mul-1-neg 21.5%

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

   9. Simplified 21.5%

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

4. Final simplification 23.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5.2 \cdot 10^{-296}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right)\right) \cdot \left(A + \left(A + C\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;F \leq 1.55 \cdot 10^{+24}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{B \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{\frac{F}{B}}\\ \end{array} \]

Alternative 11: 30.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ t_1 := 2 \cdot \left(F \cdot t_0\right)\\ \mathbf{if}\;C \leq -2.8 \cdot 10^{-34}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(A \cdot \left(2 + -0.5 \cdot \frac{B \cdot B}{C \cdot C}\right)\right)}}{t_0}\\ \mathbf{elif}\;C \leq 1060000000000:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{\frac{F}{B}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(\left(A + C\right) + \left(C + \left(0.5 \cdot \frac{B \cdot B}{C} - A\right)\right)\right)}}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* A C)))) (t_1 (* 2.0 (* F t_0))))
   (if (<= C -2.8e-34)
     (/ (- (sqrt (* t_1 (* A (+ 2.0 (* -0.5 (/ (* B B) (* C C)))))))) t_0)
     (if (<= C 1060000000000.0)
       (- (* (sqrt 2.0) (sqrt (/ F B))))
       (/
        (- (sqrt (* t_1 (+ (+ A C) (+ C (- (* 0.5 (/ (* B B) C)) A))))))
        t_0)))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double t_1 = 2.0 * (F * t_0);
	double tmp;
	if (C <= -2.8e-34) {
		tmp = -sqrt((t_1 * (A * (2.0 + (-0.5 * ((B * B) / (C * C))))))) / t_0;
	} else if (C <= 1060000000000.0) {
		tmp = -(sqrt(2.0) * sqrt((F / B)));
	} else {
		tmp = -sqrt((t_1 * ((A + C) + (C + ((0.5 * ((B * B) / C)) - A))))) / t_0;
	}
	return tmp;
}

Fortran

NOTE: B should be positive 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 = 2.0d0 * (f * t_0)
    if (c <= (-2.8d-34)) then
        tmp = -sqrt((t_1 * (a * (2.0d0 + ((-0.5d0) * ((b * b) / (c * c))))))) / t_0
    else if (c <= 1060000000000.0d0) then
        tmp = -(sqrt(2.0d0) * sqrt((f / b)))
    else
        tmp = -sqrt((t_1 * ((a + c) + (c + ((0.5d0 * ((b * b) / c)) - a))))) / t_0
    end if
    code = tmp
end function

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double t_1 = 2.0 * (F * t_0);
	double tmp;
	if (C <= -2.8e-34) {
		tmp = -Math.sqrt((t_1 * (A * (2.0 + (-0.5 * ((B * B) / (C * C))))))) / t_0;
	} else if (C <= 1060000000000.0) {
		tmp = -(Math.sqrt(2.0) * Math.sqrt((F / B)));
	} else {
		tmp = -Math.sqrt((t_1 * ((A + C) + (C + ((0.5 * ((B * B) / C)) - A))))) / t_0;
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (A * C))
	t_1 = 2.0 * (F * t_0)
	tmp = 0
	if C <= -2.8e-34:
		tmp = -math.sqrt((t_1 * (A * (2.0 + (-0.5 * ((B * B) / (C * C))))))) / t_0
	elif C <= 1060000000000.0:
		tmp = -(math.sqrt(2.0) * math.sqrt((F / B)))
	else:
		tmp = -math.sqrt((t_1 * ((A + C) + (C + ((0.5 * ((B * B) / C)) - A))))) / t_0
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	t_1 = Float64(2.0 * Float64(F * t_0))
	tmp = 0.0
	if (C <= -2.8e-34)
		tmp = Float64(Float64(-sqrt(Float64(t_1 * Float64(A * Float64(2.0 + Float64(-0.5 * Float64(Float64(B * B) / Float64(C * C)))))))) / t_0);
	elseif (C <= 1060000000000.0)
		tmp = Float64(-Float64(sqrt(2.0) * sqrt(Float64(F / B))));
	else
		tmp = Float64(Float64(-sqrt(Float64(t_1 * Float64(Float64(A + C) + Float64(C + Float64(Float64(0.5 * Float64(Float64(B * B) / C)) - A)))))) / t_0);
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (A * C));
	t_1 = 2.0 * (F * t_0);
	tmp = 0.0;
	if (C <= -2.8e-34)
		tmp = -sqrt((t_1 * (A * (2.0 + (-0.5 * ((B * B) / (C * C))))))) / t_0;
	elseif (C <= 1060000000000.0)
		tmp = -(sqrt(2.0) * sqrt((F / B)));
	else
		tmp = -sqrt((t_1 * ((A + C) + (C + ((0.5 * ((B * B) / C)) - A))))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive 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[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[C, -2.8e-34], N[((-N[Sqrt[N[(t$95$1 * N[(A * N[(2.0 + N[(-0.5 * N[(N[(B * B), $MachinePrecision] / N[(C * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[C, 1060000000000.0], (-N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), N[((-N[Sqrt[N[(t$95$1 * N[(N[(A + C), $MachinePrecision] + N[(C + N[(N[(0.5 * N[(N[(B * B), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision] - A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if C < -2.79999999999999997e-34

   1. Initial program 6.9%

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

      1. associate-*l* 6.9%

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

      2. unpow2 6.9%

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

      3. +-commutative 6.9%

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

      4. unpow2 6.9%

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

      5. associate-*l* 6.9%

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

      6. unpow2 6.9%

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

   3. Simplified 6.9%

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

   4. Taylor expanded in C around -inf 4.4%

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

      1. associate-+r+ 4.4%

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

      2. mul-1-neg 4.4%

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

      3. sub-neg 4.4%

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

      4. distribute-lft-out 4.4%

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

      5. unpow2 4.4%

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

      6. *-commutative 4.4%

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

      7. unpow2 4.4%

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

      8. times-frac 4.5%

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

      9. unpow2 4.5%

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

   6. Simplified 4.5%

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

   7. Taylor expanded in A around inf 24.6%

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

      1. unpow2 24.6%

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

      2. unpow2 24.6%

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

   9. Simplified 24.6%

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

   if -2.79999999999999997e-34 < C < 1.06e12

   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 33.3%

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

   4. Taylor expanded in C around 0 16.7%

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

      1. mul-1-neg 16.7%

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

      2. distribute-rgt-neg-in 16.7%

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

      3. unpow2 16.7%

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

      4. unpow2 16.7%

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

      5. hypot-def 28.5%

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

   6. Simplified 28.5%

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

   7. Taylor expanded in A around 0 21.1%

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

      1. associate-*r* 21.1%

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

      2. mul-1-neg 21.1%

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

   9. Simplified 21.1%

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

   if 1.06e12 < C

   1. Initial program 21.6%

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

      1. associate-*l* 21.6%

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

      2. unpow2 21.6%

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

      3. +-commutative 21.6%

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

      4. unpow2 21.6%

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

      5. associate-*l* 21.6%

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

      6. unpow2 21.6%

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

   3. Simplified 21.6%

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

   4. Taylor expanded in C around inf 30.4%

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

      1. mul-1-neg 30.4%

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

      2. unsub-neg 30.4%

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

   6. Simplified 30.5%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \color{blue}{\left(C + \left(0.5 \cdot \frac{B \cdot B + 0}{C} - A\right)\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 24.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -2.8 \cdot 10^{-34}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right)\right) \cdot \left(A \cdot \left(2 + -0.5 \cdot \frac{B \cdot B}{C \cdot C}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;C \leq 1060000000000:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{\frac{F}{B}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right)\right) \cdot \left(\left(A + C\right) + \left(C + \left(0.5 \cdot \frac{B \cdot B}{C} - A\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \end{array} \]

Alternative 12: 20.0% accurate, 4.4× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ t_1 := 2 \cdot \left(F \cdot t_0\right)\\ \mathbf{if}\;C \leq -1.8 \cdot 10^{-35}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(A \cdot \left(2 + -0.5 \cdot \frac{B \cdot B}{C \cdot C}\right)\right)}}{t_0}\\ \mathbf{elif}\;C \leq 1.55 \cdot 10^{-40}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(A + \left(A + C\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(\left(A + C\right) + \left(C + \left(0.5 \cdot \frac{B \cdot B}{C} - A\right)\right)\right)}}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* A C)))) (t_1 (* 2.0 (* F t_0))))
   (if (<= C -1.8e-35)
     (/ (- (sqrt (* t_1 (* A (+ 2.0 (* -0.5 (/ (* B B) (* C C)))))))) t_0)
     (if (<= C 1.55e-40)
       (/ (- (sqrt (* t_1 (+ A (+ A C))))) t_0)
       (/
        (- (sqrt (* t_1 (+ (+ A C) (+ C (- (* 0.5 (/ (* B B) C)) A))))))
        t_0)))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double t_1 = 2.0 * (F * t_0);
	double tmp;
	if (C <= -1.8e-35) {
		tmp = -sqrt((t_1 * (A * (2.0 + (-0.5 * ((B * B) / (C * C))))))) / t_0;
	} else if (C <= 1.55e-40) {
		tmp = -sqrt((t_1 * (A + (A + C)))) / t_0;
	} else {
		tmp = -sqrt((t_1 * ((A + C) + (C + ((0.5 * ((B * B) / C)) - A))))) / t_0;
	}
	return tmp;
}

Fortran

NOTE: B should be positive 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 = 2.0d0 * (f * t_0)
    if (c <= (-1.8d-35)) then
        tmp = -sqrt((t_1 * (a * (2.0d0 + ((-0.5d0) * ((b * b) / (c * c))))))) / t_0
    else if (c <= 1.55d-40) then
        tmp = -sqrt((t_1 * (a + (a + c)))) / t_0
    else
        tmp = -sqrt((t_1 * ((a + c) + (c + ((0.5d0 * ((b * b) / c)) - a))))) / t_0
    end if
    code = tmp
end function

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double t_1 = 2.0 * (F * t_0);
	double tmp;
	if (C <= -1.8e-35) {
		tmp = -Math.sqrt((t_1 * (A * (2.0 + (-0.5 * ((B * B) / (C * C))))))) / t_0;
	} else if (C <= 1.55e-40) {
		tmp = -Math.sqrt((t_1 * (A + (A + C)))) / t_0;
	} else {
		tmp = -Math.sqrt((t_1 * ((A + C) + (C + ((0.5 * ((B * B) / C)) - A))))) / t_0;
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (A * C))
	t_1 = 2.0 * (F * t_0)
	tmp = 0
	if C <= -1.8e-35:
		tmp = -math.sqrt((t_1 * (A * (2.0 + (-0.5 * ((B * B) / (C * C))))))) / t_0
	elif C <= 1.55e-40:
		tmp = -math.sqrt((t_1 * (A + (A + C)))) / t_0
	else:
		tmp = -math.sqrt((t_1 * ((A + C) + (C + ((0.5 * ((B * B) / C)) - A))))) / t_0
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	t_1 = Float64(2.0 * Float64(F * t_0))
	tmp = 0.0
	if (C <= -1.8e-35)
		tmp = Float64(Float64(-sqrt(Float64(t_1 * Float64(A * Float64(2.0 + Float64(-0.5 * Float64(Float64(B * B) / Float64(C * C)))))))) / t_0);
	elseif (C <= 1.55e-40)
		tmp = Float64(Float64(-sqrt(Float64(t_1 * Float64(A + Float64(A + C))))) / t_0);
	else
		tmp = Float64(Float64(-sqrt(Float64(t_1 * Float64(Float64(A + C) + Float64(C + Float64(Float64(0.5 * Float64(Float64(B * B) / C)) - A)))))) / t_0);
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (A * C));
	t_1 = 2.0 * (F * t_0);
	tmp = 0.0;
	if (C <= -1.8e-35)
		tmp = -sqrt((t_1 * (A * (2.0 + (-0.5 * ((B * B) / (C * C))))))) / t_0;
	elseif (C <= 1.55e-40)
		tmp = -sqrt((t_1 * (A + (A + C)))) / t_0;
	else
		tmp = -sqrt((t_1 * ((A + C) + (C + ((0.5 * ((B * B) / C)) - A))))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive 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[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[C, -1.8e-35], N[((-N[Sqrt[N[(t$95$1 * N[(A * N[(2.0 + N[(-0.5 * N[(N[(B * B), $MachinePrecision] / N[(C * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[C, 1.55e-40], N[((-N[Sqrt[N[(t$95$1 * N[(A + N[(A + C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[((-N[Sqrt[N[(t$95$1 * N[(N[(A + C), $MachinePrecision] + N[(C + N[(N[(0.5 * N[(N[(B * B), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision] - A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if C < -1.80000000000000009e-35

   1. Initial program 6.9%

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

      1. associate-*l* 6.9%

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

      2. unpow2 6.9%

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

      3. +-commutative 6.9%

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

      4. unpow2 6.9%

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

      5. associate-*l* 6.9%

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

      6. unpow2 6.9%

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

   3. Simplified 6.9%

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

   4. Taylor expanded in C around -inf 4.4%

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

      1. associate-+r+ 4.4%

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

      2. mul-1-neg 4.4%

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

      3. sub-neg 4.4%

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

      4. distribute-lft-out 4.4%

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

      5. unpow2 4.4%

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

      6. *-commutative 4.4%

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

      7. unpow2 4.4%

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

      8. times-frac 4.5%

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

      9. unpow2 4.5%

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

   6. Simplified 4.5%

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

   7. Taylor expanded in A around inf 24.6%

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

      1. unpow2 24.6%

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

      2. unpow2 24.6%

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

   9. Simplified 24.6%

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

   if -1.80000000000000009e-35 < C < 1.55000000000000005e-40

   1. Initial program 28.9%

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

      1. associate-*l* 28.9%

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

      2. unpow2 28.9%

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

      3. +-commutative 28.9%

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

      4. unpow2 28.9%

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

      5. associate-*l* 28.9%

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

      6. unpow2 28.9%

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

   3. Simplified 28.9%

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

   4. Taylor expanded in A around inf 17.1%

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

   if 1.55000000000000005e-40 < C

   1. Initial program 19.3%

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

      1. associate-*l* 19.3%

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

      2. unpow2 19.3%

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

      3. +-commutative 19.3%

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

      4. unpow2 19.3%

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

      5. associate-*l* 19.3%

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

      6. unpow2 19.3%

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

   3. Simplified 19.3%

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

   4. Taylor expanded in C around inf 26.3%

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

      1. mul-1-neg 26.3%

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

      2. unsub-neg 26.3%

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

   6. Simplified 26.6%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \color{blue}{\left(C + \left(0.5 \cdot \frac{B \cdot B + 0}{C} - A\right)\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 21.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -1.8 \cdot 10^{-35}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right)\right) \cdot \left(A \cdot \left(2 + -0.5 \cdot \frac{B \cdot B}{C \cdot C}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;C \leq 1.55 \cdot 10^{-40}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right)\right) \cdot \left(A + \left(A + C\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right)\right) \cdot \left(\left(A + C\right) + \left(C + \left(0.5 \cdot \frac{B \cdot B}{C} - A\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \end{array} \]

Alternative 13: 20.0% accurate, 4.5× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ t_1 := 2 \cdot \left(F \cdot t_0\right)\\ \mathbf{if}\;C \leq -8 \cdot 10^{-42}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(A \cdot \left(2 + -0.5 \cdot \frac{B \cdot B}{C \cdot C}\right)\right)}}{t_0}\\ \mathbf{elif}\;C \leq 8.2 \cdot 10^{-41}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(A + \left(A + C\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(\left(A + C\right) + \left(C - A\right)\right)}}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* A C)))) (t_1 (* 2.0 (* F t_0))))
   (if (<= C -8e-42)
     (/ (- (sqrt (* t_1 (* A (+ 2.0 (* -0.5 (/ (* B B) (* C C)))))))) t_0)
     (if (<= C 8.2e-41)
       (/ (- (sqrt (* t_1 (+ A (+ A C))))) t_0)
       (/ (- (sqrt (* t_1 (+ (+ A C) (- C A))))) t_0)))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double t_1 = 2.0 * (F * t_0);
	double tmp;
	if (C <= -8e-42) {
		tmp = -sqrt((t_1 * (A * (2.0 + (-0.5 * ((B * B) / (C * C))))))) / t_0;
	} else if (C <= 8.2e-41) {
		tmp = -sqrt((t_1 * (A + (A + C)))) / t_0;
	} else {
		tmp = -sqrt((t_1 * ((A + C) + (C - A)))) / t_0;
	}
	return tmp;
}

Fortran

NOTE: B should be positive 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 = 2.0d0 * (f * t_0)
    if (c <= (-8d-42)) then
        tmp = -sqrt((t_1 * (a * (2.0d0 + ((-0.5d0) * ((b * b) / (c * c))))))) / t_0
    else if (c <= 8.2d-41) then
        tmp = -sqrt((t_1 * (a + (a + c)))) / t_0
    else
        tmp = -sqrt((t_1 * ((a + c) + (c - a)))) / t_0
    end if
    code = tmp
end function

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double t_1 = 2.0 * (F * t_0);
	double tmp;
	if (C <= -8e-42) {
		tmp = -Math.sqrt((t_1 * (A * (2.0 + (-0.5 * ((B * B) / (C * C))))))) / t_0;
	} else if (C <= 8.2e-41) {
		tmp = -Math.sqrt((t_1 * (A + (A + C)))) / t_0;
	} else {
		tmp = -Math.sqrt((t_1 * ((A + C) + (C - A)))) / t_0;
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (A * C))
	t_1 = 2.0 * (F * t_0)
	tmp = 0
	if C <= -8e-42:
		tmp = -math.sqrt((t_1 * (A * (2.0 + (-0.5 * ((B * B) / (C * C))))))) / t_0
	elif C <= 8.2e-41:
		tmp = -math.sqrt((t_1 * (A + (A + C)))) / t_0
	else:
		tmp = -math.sqrt((t_1 * ((A + C) + (C - A)))) / t_0
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	t_1 = Float64(2.0 * Float64(F * t_0))
	tmp = 0.0
	if (C <= -8e-42)
		tmp = Float64(Float64(-sqrt(Float64(t_1 * Float64(A * Float64(2.0 + Float64(-0.5 * Float64(Float64(B * B) / Float64(C * C)))))))) / t_0);
	elseif (C <= 8.2e-41)
		tmp = Float64(Float64(-sqrt(Float64(t_1 * Float64(A + Float64(A + C))))) / t_0);
	else
		tmp = Float64(Float64(-sqrt(Float64(t_1 * Float64(Float64(A + C) + Float64(C - A))))) / t_0);
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (A * C));
	t_1 = 2.0 * (F * t_0);
	tmp = 0.0;
	if (C <= -8e-42)
		tmp = -sqrt((t_1 * (A * (2.0 + (-0.5 * ((B * B) / (C * C))))))) / t_0;
	elseif (C <= 8.2e-41)
		tmp = -sqrt((t_1 * (A + (A + C)))) / t_0;
	else
		tmp = -sqrt((t_1 * ((A + C) + (C - A)))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive 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[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[C, -8e-42], N[((-N[Sqrt[N[(t$95$1 * N[(A * N[(2.0 + N[(-0.5 * N[(N[(B * B), $MachinePrecision] / N[(C * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[C, 8.2e-41], N[((-N[Sqrt[N[(t$95$1 * N[(A + N[(A + C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[((-N[Sqrt[N[(t$95$1 * N[(N[(A + C), $MachinePrecision] + N[(C - A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if C < -8.0000000000000003e-42

   1. Initial program 6.9%

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

      1. associate-*l* 6.9%

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

      2. unpow2 6.9%

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

      3. +-commutative 6.9%

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

      4. unpow2 6.9%

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

      5. associate-*l* 6.9%

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

      6. unpow2 6.9%

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

   3. Simplified 6.9%

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

   4. Taylor expanded in C around -inf 4.4%

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

      1. associate-+r+ 4.4%

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

      2. mul-1-neg 4.4%

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

      3. sub-neg 4.4%

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

      4. distribute-lft-out 4.4%

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

      5. unpow2 4.4%

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

      6. *-commutative 4.4%

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

      7. unpow2 4.4%

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

      8. times-frac 4.5%

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

      9. unpow2 4.5%

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

   6. Simplified 4.5%

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

   7. Taylor expanded in A around inf 24.6%

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

      1. unpow2 24.6%

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

      2. unpow2 24.6%

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

   9. Simplified 24.6%

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

   if -8.0000000000000003e-42 < C < 8.20000000000000028e-41

   1. Initial program 28.9%

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

      1. associate-*l* 28.9%

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

      2. unpow2 28.9%

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

      3. +-commutative 28.9%

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

      4. unpow2 28.9%

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

      5. associate-*l* 28.9%

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

      6. unpow2 28.9%

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

   3. Simplified 28.9%

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

   4. Taylor expanded in A around inf 17.1%

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

   if 8.20000000000000028e-41 < C

   1. Initial program 19.3%

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

      1. associate-*l* 19.3%

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

      2. unpow2 19.3%

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

      3. +-commutative 19.3%

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

      4. unpow2 19.3%

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

      5. associate-*l* 19.3%

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

      6. unpow2 19.3%

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

   3. Simplified 19.3%

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

   4. Taylor expanded in A around -inf 26.5%

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

      1. mul-1-neg 26.5%

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

      2. sub-neg 26.5%

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

   6. Simplified 26.5%

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

4. Final simplification 21.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -8 \cdot 10^{-42}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right)\right) \cdot \left(A \cdot \left(2 + -0.5 \cdot \frac{B \cdot B}{C \cdot C}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;C \leq 8.2 \cdot 10^{-41}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right)\right) \cdot \left(A + \left(A + C\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right)\right) \cdot \left(\left(A + C\right) + \left(C - A\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \end{array} \]

Alternative 14: 14.7% accurate, 4.7× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ t_1 := 2 \cdot \left(F \cdot t_0\right)\\ \mathbf{if}\;C \leq 2.4 \cdot 10^{-41}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(\left(A + C\right) + \left(A - C\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(\left(A + C\right) + \left(C - A\right)\right)}}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* A C)))) (t_1 (* 2.0 (* F t_0))))
   (if (<= C 2.4e-41)
     (/ (- (sqrt (* t_1 (+ (+ A C) (- A C))))) t_0)
     (/ (- (sqrt (* t_1 (+ (+ A C) (- C A))))) t_0))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double t_1 = 2.0 * (F * t_0);
	double tmp;
	if (C <= 2.4e-41) {
		tmp = -sqrt((t_1 * ((A + C) + (A - C)))) / t_0;
	} else {
		tmp = -sqrt((t_1 * ((A + C) + (C - A)))) / t_0;
	}
	return tmp;
}

Fortran

NOTE: B should be positive 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 = 2.0d0 * (f * t_0)
    if (c <= 2.4d-41) then
        tmp = -sqrt((t_1 * ((a + c) + (a - c)))) / t_0
    else
        tmp = -sqrt((t_1 * ((a + c) + (c - a)))) / t_0
    end if
    code = tmp
end function

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double t_1 = 2.0 * (F * t_0);
	double tmp;
	if (C <= 2.4e-41) {
		tmp = -Math.sqrt((t_1 * ((A + C) + (A - C)))) / t_0;
	} else {
		tmp = -Math.sqrt((t_1 * ((A + C) + (C - A)))) / t_0;
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (A * C))
	t_1 = 2.0 * (F * t_0)
	tmp = 0
	if C <= 2.4e-41:
		tmp = -math.sqrt((t_1 * ((A + C) + (A - C)))) / t_0
	else:
		tmp = -math.sqrt((t_1 * ((A + C) + (C - A)))) / t_0
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	t_1 = Float64(2.0 * Float64(F * t_0))
	tmp = 0.0
	if (C <= 2.4e-41)
		tmp = Float64(Float64(-sqrt(Float64(t_1 * Float64(Float64(A + C) + Float64(A - C))))) / t_0);
	else
		tmp = Float64(Float64(-sqrt(Float64(t_1 * Float64(Float64(A + C) + Float64(C - A))))) / t_0);
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (A * C));
	t_1 = 2.0 * (F * t_0);
	tmp = 0.0;
	if (C <= 2.4e-41)
		tmp = -sqrt((t_1 * ((A + C) + (A - C)))) / t_0;
	else
		tmp = -sqrt((t_1 * ((A + C) + (C - A)))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive 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[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[C, 2.4e-41], N[((-N[Sqrt[N[(t$95$1 * N[(N[(A + C), $MachinePrecision] + N[(A - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[((-N[Sqrt[N[(t$95$1 * N[(N[(A + C), $MachinePrecision] + N[(C - A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if C < 2.40000000000000022e-41

   1. Initial program 20.9%

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

      1. associate-*l* 20.9%

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

      2. unpow2 20.9%

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

      3. +-commutative 20.9%

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

      4. unpow2 20.9%

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

      5. associate-*l* 20.9%

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

      6. unpow2 20.9%

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

   3. Simplified 20.9%

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

   4. Taylor expanded in B around 0 12.4%

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

   if 2.40000000000000022e-41 < C

   1. Initial program 19.3%

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

      1. associate-*l* 19.3%

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

      2. unpow2 19.3%

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

      3. +-commutative 19.3%

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

      4. unpow2 19.3%

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

      5. associate-*l* 19.3%

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

      6. unpow2 19.3%

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

   3. Simplified 19.3%

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

   4. Taylor expanded in A around -inf 26.5%

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

      1. mul-1-neg 26.5%

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

      2. sub-neg 26.5%

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

   6. Simplified 26.5%

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

4. Final simplification 16.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 2.4 \cdot 10^{-41}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right)\right) \cdot \left(\left(A + C\right) + \left(A - C\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right)\right) \cdot \left(\left(A + C\right) + \left(C - A\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \end{array} \]

Alternative 15: 11.4% accurate, 4.8× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq 1.32 \cdot 10^{+85}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot t_0\right)\right) \cdot \left(A + \left(A + C\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-\frac{2}{B} \cdot \sqrt{A \cdot F}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* A C)))))
   (if (<= B 1.32e+85)
     (/ (- (sqrt (* (* 2.0 (* F t_0)) (+ A (+ A C))))) t_0)
     (- (* (/ 2.0 B) (sqrt (* A F)))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (B <= 1.32e+85) {
		tmp = -sqrt(((2.0 * (F * t_0)) * (A + (A + C)))) / t_0;
	} else {
		tmp = -((2.0 / B) * sqrt((A * F)));
	}
	return tmp;
}

Fortran

NOTE: B should be positive 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 <= 1.32d+85) then
        tmp = -sqrt(((2.0d0 * (f * t_0)) * (a + (a + c)))) / t_0
    else
        tmp = -((2.0d0 / b) * sqrt((a * f)))
    end if
    code = tmp
end function

Java

B = Math.abs(B);
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 <= 1.32e+85) {
		tmp = -Math.sqrt(((2.0 * (F * t_0)) * (A + (A + C)))) / t_0;
	} else {
		tmp = -((2.0 / B) * Math.sqrt((A * F)));
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (A * C))
	tmp = 0
	if B <= 1.32e+85:
		tmp = -math.sqrt(((2.0 * (F * t_0)) * (A + (A + C)))) / t_0
	else:
		tmp = -((2.0 / B) * math.sqrt((A * F)))
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	tmp = 0.0
	if (B <= 1.32e+85)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(F * t_0)) * Float64(A + Float64(A + C))))) / t_0);
	else
		tmp = Float64(-Float64(Float64(2.0 / B) * sqrt(Float64(A * F))));
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (A * C));
	tmp = 0.0;
	if (B <= 1.32e+85)
		tmp = -sqrt(((2.0 * (F * t_0)) * (A + (A + C)))) / t_0;
	else
		tmp = -((2.0 / B) * sqrt((A * F)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive 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, 1.32e+85], N[((-N[Sqrt[N[(N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(A + N[(A + C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], (-N[(N[(2.0 / B), $MachinePrecision] * N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]]

Derivation

1. Split input into 2 regimes

2. if B < 1.32000000000000007e85

   1. Initial program 24.0%

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

      1. associate-*l* 24.0%

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

      2. unpow2 24.0%

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

      3. +-commutative 24.0%

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

      4. unpow2 24.0%

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

      5. associate-*l* 24.0%

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

      6. unpow2 24.0%

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

   3. Simplified 24.0%

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

   4. Taylor expanded in A around inf 12.7%

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

   if 1.32000000000000007e85 < B

   1. Initial program 2.9%

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

   3. Simplified 2.9%

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

   4. Taylor expanded in C around 0 9.3%

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

      1. mul-1-neg 9.3%

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

      2. distribute-rgt-neg-in 9.3%

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

      3. unpow2 9.3%

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

      4. unpow2 9.3%

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

      5. hypot-def 55.1%

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

   6. Simplified 55.1%

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

   7. Taylor expanded in B around 0 3.2%

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

      1. associate-*r* 3.2%

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

      2. mul-1-neg 3.2%

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

      3. unpow2 3.2%

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

      4. rem-square-sqrt 3.2%

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

   9. Simplified 3.2%

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

4. Final simplification 11.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.32 \cdot 10^{+85}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right)\right) \cdot \left(A + \left(A + C\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{2}{B} \cdot \sqrt{A \cdot F}\\ \end{array} \]

Alternative 16: 7.2% accurate, 5.0× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;A \leq 10^{-197}:\\ \;\;\;\;\frac{-\sqrt{C \cdot \left(2 \cdot \left(F \cdot t_0\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-\frac{2}{B} \cdot \sqrt{A \cdot F}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* A C)))))
   (if (<= A 1e-197)
     (/ (- (sqrt (* C (* 2.0 (* F t_0))))) t_0)
     (- (* (/ 2.0 B) (sqrt (* A F)))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (A <= 1e-197) {
		tmp = -sqrt((C * (2.0 * (F * t_0)))) / t_0;
	} else {
		tmp = -((2.0 / B) * sqrt((A * F)));
	}
	return tmp;
}

Fortran

NOTE: B should be positive 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 (a <= 1d-197) then
        tmp = -sqrt((c * (2.0d0 * (f * t_0)))) / t_0
    else
        tmp = -((2.0d0 / b) * sqrt((a * f)))
    end if
    code = tmp
end function

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (A <= 1e-197) {
		tmp = -Math.sqrt((C * (2.0 * (F * t_0)))) / t_0;
	} else {
		tmp = -((2.0 / B) * Math.sqrt((A * F)));
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (A * C))
	tmp = 0
	if A <= 1e-197:
		tmp = -math.sqrt((C * (2.0 * (F * t_0)))) / t_0
	else:
		tmp = -((2.0 / B) * math.sqrt((A * F)))
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	tmp = 0.0
	if (A <= 1e-197)
		tmp = Float64(Float64(-sqrt(Float64(C * Float64(2.0 * Float64(F * t_0))))) / t_0);
	else
		tmp = Float64(-Float64(Float64(2.0 / B) * sqrt(Float64(A * F))));
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (A * C));
	tmp = 0.0;
	if (A <= 1e-197)
		tmp = -sqrt((C * (2.0 * (F * t_0)))) / t_0;
	else
		tmp = -((2.0 / B) * sqrt((A * F)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive 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[A, 1e-197], N[((-N[Sqrt[N[(C * N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], (-N[(N[(2.0 / B), $MachinePrecision] * N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]]

Derivation

1. Split input into 2 regimes

2. if A < 9.9999999999999999e-198

   1. Initial program 18.5%

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

      1. associate-*l* 18.5%

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

      2. unpow2 18.5%

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

      3. +-commutative 18.5%

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

      4. unpow2 18.5%

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

      5. associate-*l* 18.5%

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

      6. unpow2 18.5%

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

   3. Simplified 18.5%

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

   4. Taylor expanded in A around -inf 4.3%

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

      1. mul-1-neg 4.3%

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

   6. Simplified 4.3%

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

   7. Taylor expanded in A around 0 6.7%

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

   if 9.9999999999999999e-198 < A

   1. Initial program 23.4%

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

   3. Simplified 30.9%

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

   4. Taylor expanded in C around 0 9.9%

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

      1. mul-1-neg 9.9%

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

      2. distribute-rgt-neg-in 9.9%

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

      3. unpow2 9.9%

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

      4. unpow2 9.9%

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

      5. hypot-def 20.6%

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

   6. Simplified 20.6%

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

   7. Taylor expanded in B around 0 7.3%

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

      1. associate-*r* 7.3%

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

      2. mul-1-neg 7.3%

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

      3. unpow2 7.3%

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

      4. rem-square-sqrt 7.3%

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

   9. Simplified 7.3%

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

4. Final simplification 7.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq 10^{-197}:\\ \;\;\;\;\frac{-\sqrt{C \cdot \left(2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{2}{B} \cdot \sqrt{A \cdot F}\\ \end{array} \]

Alternative 17: 7.1% accurate, 5.2× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} \mathbf{if}\;A \leq 1.2 \cdot 10^{-235}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -8\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{2}{B} \cdot \sqrt{A \cdot F}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (if (<= A 1.2e-235)
   (/ (- (sqrt (* (* A -8.0) (* F (* C C))))) (- (* B B) (* 4.0 (* A C))))
   (- (* (/ 2.0 B) (sqrt (* A F))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double tmp;
	if (A <= 1.2e-235) {
		tmp = -sqrt(((A * -8.0) * (F * (C * C)))) / ((B * B) - (4.0 * (A * C)));
	} else {
		tmp = -((2.0 / B) * sqrt((A * F)));
	}
	return tmp;
}

Fortran

NOTE: B should be positive before calling this function
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (a <= 1.2d-235) then
        tmp = -sqrt(((a * (-8.0d0)) * (f * (c * c)))) / ((b * b) - (4.0d0 * (a * c)))
    else
        tmp = -((2.0d0 / b) * sqrt((a * f)))
    end if
    code = tmp
end function

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	double tmp;
	if (A <= 1.2e-235) {
		tmp = -Math.sqrt(((A * -8.0) * (F * (C * C)))) / ((B * B) - (4.0 * (A * C)));
	} else {
		tmp = -((2.0 / B) * Math.sqrt((A * F)));
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	tmp = 0
	if A <= 1.2e-235:
		tmp = -math.sqrt(((A * -8.0) * (F * (C * C)))) / ((B * B) - (4.0 * (A * C)))
	else:
		tmp = -((2.0 / B) * math.sqrt((A * F)))
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	tmp = 0.0
	if (A <= 1.2e-235)
		tmp = Float64(Float64(-sqrt(Float64(Float64(A * -8.0) * Float64(F * Float64(C * C))))) / Float64(Float64(B * B) - Float64(4.0 * Float64(A * C))));
	else
		tmp = Float64(-Float64(Float64(2.0 / B) * sqrt(Float64(A * F))));
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (A <= 1.2e-235)
		tmp = -sqrt(((A * -8.0) * (F * (C * C)))) / ((B * B) - (4.0 * (A * C)));
	else
		tmp = -((2.0 / B) * sqrt((A * F)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := If[LessEqual[A, 1.2e-235], N[((-N[Sqrt[N[(N[(A * -8.0), $MachinePrecision] * N[(F * N[(C * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(N[(2.0 / B), $MachinePrecision] * N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if A < 1.20000000000000005e-235

   1. Initial program 17.7%

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

      1. associate-*l* 17.7%

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

      2. unpow2 17.7%

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

      3. +-commutative 17.7%

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

      4. unpow2 17.7%

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

      5. associate-*l* 17.7%

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

      6. unpow2 17.7%

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

   3. Simplified 17.7%

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

   4. Taylor expanded in A around -inf 4.2%

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

      1. mul-1-neg 4.2%

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

   6. Simplified 4.2%

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

   7. Taylor expanded in B around 0 5.9%

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

      1. associate-*r* 5.9%

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

      2. *-commutative 5.9%

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

      3. unpow2 5.9%

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

   9. Simplified 5.9%

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

   if 1.20000000000000005e-235 < A

   1. Initial program 24.3%

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

   3. Simplified 32.3%

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

   4. Taylor expanded in C around 0 10.5%

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

      1. mul-1-neg 10.5%

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

      2. distribute-rgt-neg-in 10.5%

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

      3. unpow2 10.5%

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

      4. unpow2 10.5%

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

      5. hypot-def 20.6%

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

   6. Simplified 20.6%

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

   7. Taylor expanded in B around 0 7.1%

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

      1. associate-*r* 7.1%

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

      2. mul-1-neg 7.1%

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

      3. unpow2 7.1%

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

      4. rem-square-sqrt 7.1%

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

   9. Simplified 7.1%

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

4. Final simplification 6.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq 1.2 \cdot 10^{-235}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -8\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{2}{B} \cdot \sqrt{A \cdot F}\\ \end{array} \]

Alternative 18: 4.8% accurate, 5.9× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ -\frac{2}{B} \cdot \sqrt{A \cdot F} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F) :precision binary64 (- (* (/ 2.0 B) (sqrt (* A F)))))

C

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

Fortran

NOTE: B should be positive 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 / b) * sqrt((a * f)))
end function

Java

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

Python

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

Julia

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

MATLAB

B = abs(B)
function tmp = code(A, B, C, F)
	tmp = -((2.0 / B) * sqrt((A * F)));
end

Wolfram

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

Derivation

1. Initial program 20.4%

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

3. Simplified 27.2%

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

4. Taylor expanded in C around 0 10.6%

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

   1. mul-1-neg 10.6%

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

   2. distribute-rgt-neg-in 10.6%

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

   3. unpow2 10.6%

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

   4. unpow2 10.6%

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

   5. hypot-def 19.1%

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

6. Simplified 19.1%

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

7. Taylor expanded in B around 0 3.3%

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

   1. associate-*r* 3.3%

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

   2. mul-1-neg 3.3%

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

   3. unpow2 3.3%

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

   4. rem-square-sqrt 3.3%

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

9. Simplified 3.3%

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

10. Final simplification 3.3%

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

Reproduce

herbie shell --seed 2023208 
(FPCore (A B C F)
  :name "ABCF->ab-angle a"
  :precision binary64
  (/ (- (sqrt (* (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F)) (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))) (- (pow B 2.0) (* (* 4.0 A) C))))