ABCF->ab-angle b

Percentage Accurate: 18.7% → 41.9%

Time: 27.3s

Alternatives: 17

Speedup: 5.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 18.7% 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: 41.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;{B}^{2} \leq 200000000000:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(t_0 \cdot F\right) \cdot \left(2 \cdot A\right)\right)}}{t_0}\\ \mathbf{elif}\;{B}^{2} \leq 10^{+308}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \left(-\sqrt{2 \cdot \left(F \cdot \left(A + \left(C - \mathsf{hypot}\left(A - C, B\right)\right)\right)\right)}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* A C)))))
   (if (<= (pow B 2.0) 200000000000.0)
     (/ (- (sqrt (* 2.0 (* (* t_0 F) (* 2.0 A))))) t_0)
     (if (<= (pow B 2.0) 1e+308)
       (/
        (*
         (sqrt (fma B B (* C (* A -4.0))))
         (- (sqrt (* 2.0 (* F (+ A (- C (hypot (- A C) B))))))))
        (fma B B (* A (* C -4.0))))
       (* (sqrt (* F (- A (hypot A B)))) (- (/ (sqrt 2.0) B)))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (pow(B, 2.0) <= 200000000000.0) {
		tmp = -sqrt((2.0 * ((t_0 * F) * (2.0 * A)))) / t_0;
	} else if (pow(B, 2.0) <= 1e+308) {
		tmp = (sqrt(fma(B, B, (C * (A * -4.0)))) * -sqrt((2.0 * (F * (A + (C - hypot((A - C), B))))))) / fma(B, B, (A * (C * -4.0)));
	} else {
		tmp = sqrt((F * (A - hypot(A, B)))) * -(sqrt(2.0) / B);
	}
	return tmp;
}

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	tmp = 0.0
	if ((B ^ 2.0) <= 200000000000.0)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(t_0 * F) * Float64(2.0 * A))))) / t_0);
	elseif ((B ^ 2.0) <= 1e+308)
		tmp = Float64(Float64(sqrt(fma(B, B, Float64(C * Float64(A * -4.0)))) * Float64(-sqrt(Float64(2.0 * Float64(F * Float64(A + Float64(C - hypot(Float64(A - C), B)))))))) / fma(B, B, Float64(A * Float64(C * -4.0))));
	else
		tmp = Float64(sqrt(Float64(F * Float64(A - hypot(A, B)))) * Float64(-Float64(sqrt(2.0) / B)));
	end
	return tmp
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 200000000000.0], N[((-N[Sqrt[N[(2.0 * N[(N[(t$95$0 * F), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 1e+308], N[(N[(N[Sqrt[N[(B * B + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * N[(F * N[(A + N[(C - N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / N[(B * B + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F * N[(A - N[Sqrt[A ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B 2) < 2e11

   1. Initial program 26.5%

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

   3. Simplified 26.4%

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

   4. Taylor expanded in A around -inf 30.3%

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

      1. *-commutative 30.3%

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

   6. Simplified 30.3%

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

   if 2e11 < (pow.f64 B 2) < 1e308

   1. Initial program 27.2%

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

   3. Simplified 32.1%

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

      1. sqrt-prod 45.0%

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

      2. associate-*r* 45.0%

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

      3. *-commutative 45.0%

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

      4. associate-*l* 45.0%

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

      5. associate--r- 45.0%

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

      6. +-commutative 45.0%

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

   5. Applied egg-rr 45.0%

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

      1. hypot-def 38.9%

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

      2. unpow2 38.9%

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

      3. unpow2 38.9%

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

      4. +-commutative 38.9%

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

      5. unpow2 38.9%

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

      6. unpow2 38.9%

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

      7. hypot-def 45.0%

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

   7. Simplified 45.0%

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

   if 1e308 < (pow.f64 B 2)

   1. Initial program 1.6%

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

   3. Simplified 1.6%

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

   4. Taylor expanded in C around 0 3.2%

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

      1. mul-1-neg 3.2%

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

      2. *-commutative 3.2%

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

      3. +-commutative 3.2%

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

      4. unpow2 3.2%

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

      5. unpow2 3.2%

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

      6. hypot-def 34.7%

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

   6. Simplified 34.7%

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

4. Final simplification 35.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 200000000000:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(2 \cdot A\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;{B}^{2} \leq 10^{+308}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \left(-\sqrt{2 \cdot \left(F \cdot \left(A + \left(C - \mathsf{hypot}\left(A - C, B\right)\right)\right)\right)}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \end{array} \]

Alternative 2: 40.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_1 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -7.4 \cdot 10^{+101}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, C \cdot F, F \cdot \left(4 \cdot A\right)\right)}}{B}\\ \mathbf{elif}\;B \leq -5.2 \cdot 10^{-19}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot t_0\right)\right) \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 400000:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(t_1 \cdot F\right) \cdot \left(2 \cdot A\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma B B (* C (* A -4.0)))) (t_1 (- (* B B) (* 4.0 (* A C)))))
   (if (<= B -7.4e+101)
     (/ (sqrt (fma -4.0 (* C F) (* F (* 4.0 A)))) B)
     (if (<= B -5.2e-19)
       (/ (- (sqrt (* (* 2.0 (* F t_0)) (+ A (- C (hypot B (- A C))))))) t_0)
       (if (<= B 400000.0)
         (/ (- (sqrt (* 2.0 (* (* t_1 F) (* 2.0 A))))) t_1)
         (* (sqrt (* F (- A (hypot A B)))) (- (/ (sqrt 2.0) B))))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = fma(B, B, (C * (A * -4.0)));
	double t_1 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (B <= -7.4e+101) {
		tmp = sqrt(fma(-4.0, (C * F), (F * (4.0 * A)))) / B;
	} else if (B <= -5.2e-19) {
		tmp = -sqrt(((2.0 * (F * t_0)) * (A + (C - hypot(B, (A - C)))))) / t_0;
	} else if (B <= 400000.0) {
		tmp = -sqrt((2.0 * ((t_1 * F) * (2.0 * A)))) / t_1;
	} else {
		tmp = sqrt((F * (A - hypot(A, B)))) * -(sqrt(2.0) / B);
	}
	return tmp;
}

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = fma(B, B, Float64(C * Float64(A * -4.0)))
	t_1 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	tmp = 0.0
	if (B <= -7.4e+101)
		tmp = Float64(sqrt(fma(-4.0, Float64(C * F), Float64(F * Float64(4.0 * A)))) / B);
	elseif (B <= -5.2e-19)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(F * t_0)) * Float64(A + Float64(C - hypot(B, Float64(A - C))))))) / t_0);
	elseif (B <= 400000.0)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(t_1 * F) * Float64(2.0 * A))))) / t_1);
	else
		tmp = Float64(sqrt(Float64(F * Float64(A - hypot(A, B)))) * Float64(-Float64(sqrt(2.0) / B)));
	end
	return tmp
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(B * B + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -7.4e+101], N[(N[Sqrt[N[(-4.0 * N[(C * F), $MachinePrecision] + N[(F * N[(4.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision], If[LessEqual[B, -5.2e-19], N[((-N[Sqrt[N[(N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(A + N[(C - N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[B, 400000.0], N[((-N[Sqrt[N[(2.0 * N[(N[(t$95$1 * F), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], N[(N[Sqrt[N[(F * N[(A - N[Sqrt[A ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision])), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < -7.3999999999999995e101

   1. Initial program 1.1%

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

   3. Simplified 1.1%

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

   4. Taylor expanded in A around -inf 0.4%

      \[\leadsto \frac{-\sqrt{\color{blue}{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right) + \left(4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right) + -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
   5. Step-by-step derivation

      1. fma-def 0.4%

         \[\leadsto \frac{-\sqrt{\color{blue}{\mathsf{fma}\left(-16, {A}^{2} \cdot \left(C \cdot F\right), 4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right) + -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      2. unpow2 0.4%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \color{blue}{\left(A \cdot A\right)} \cdot \left(C \cdot F\right), 4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right) + -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      3. fma-def 0.4%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \color{blue}{\mathsf{fma}\left(4, A \cdot \left(F \cdot {B}^{2}\right), -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      4. *-commutative 0.4%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \color{blue}{\left({B}^{2} \cdot F\right)}, -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      5. unpow2 0.4%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\color{blue}{\left(B \cdot B\right)} \cdot F\right), -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      6. *-commutative 0.4%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\left(B \cdot B\right) \cdot F\right), -4 \cdot \left(C \cdot \color{blue}{\left({B}^{2} \cdot F\right)}\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      7. unpow2 0.4%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\left(B \cdot B\right) \cdot F\right), -4 \cdot \left(C \cdot \left(\color{blue}{\left(B \cdot B\right)} \cdot F\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   6. Simplified 0.4%

      \[\leadsto \frac{-\sqrt{\color{blue}{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\left(B \cdot B\right) \cdot F\right), -4 \cdot \left(C \cdot \left(\left(B \cdot B\right) \cdot F\right)\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   7. Taylor expanded in B around -inf 6.6%

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

      1. associate-*r/ 6.6%

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

      2. *-rgt-identity 6.6%

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

      3. fma-def 6.6%

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

      4. *-commutative 6.6%

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

      5. associate-*r* 6.6%

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

   9. Simplified 6.6%

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

   if -7.3999999999999995e101 < B < -5.20000000000000026e-19

   1. Initial program 41.8%

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

   3. Simplified 47.8%

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

   if -5.20000000000000026e-19 < B < 4e5

   1. Initial program 26.1%

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

   3. Simplified 26.0%

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

   4. Taylor expanded in A around -inf 30.7%

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

      1. *-commutative 30.7%

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

   6. Simplified 30.7%

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

   if 4e5 < B

   1. Initial program 18.3%

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

   3. Simplified 18.3%

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

   4. Taylor expanded in C around 0 23.6%

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

      1. mul-1-neg 23.6%

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

      2. *-commutative 23.6%

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

      3. +-commutative 23.6%

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

      4. unpow2 23.6%

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

      5. unpow2 23.6%

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

      6. hypot-def 54.8%

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

   6. Simplified 54.8%

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

4. Final simplification 33.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -7.4 \cdot 10^{+101}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, C \cdot F, F \cdot \left(4 \cdot A\right)\right)}}{B}\\ \mathbf{elif}\;B \leq -5.2 \cdot 10^{-19}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)\right) \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 400000:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(2 \cdot A\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \end{array} \]

Alternative 3: 40.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B + \left(A \cdot C\right) \cdot -4\\ t_1 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -7.2 \cdot 10^{+101}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, C \cdot F, F \cdot \left(4 \cdot A\right)\right)}}{B}\\ \mathbf{elif}\;B \leq -5.2 \cdot 10^{-18}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 1900000:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(t_1 \cdot F\right) \cdot \left(2 \cdot A\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (+ (* B B) (* (* A C) -4.0))) (t_1 (- (* B B) (* 4.0 (* A C)))))
   (if (<= B -7.2e+101)
     (/ (sqrt (fma -4.0 (* C F) (* F (* 4.0 A)))) B)
     (if (<= B -5.2e-18)
       (/ (- (sqrt (* 2.0 (* t_0 (* F (+ A (- C (hypot B (- A C))))))))) t_0)
       (if (<= B 1900000.0)
         (/ (- (sqrt (* 2.0 (* (* t_1 F) (* 2.0 A))))) t_1)
         (* (sqrt (* F (- A (hypot A B)))) (- (/ (sqrt 2.0) B))))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) + ((A * C) * -4.0);
	double t_1 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (B <= -7.2e+101) {
		tmp = sqrt(fma(-4.0, (C * F), (F * (4.0 * A)))) / B;
	} else if (B <= -5.2e-18) {
		tmp = -sqrt((2.0 * (t_0 * (F * (A + (C - hypot(B, (A - C)))))))) / t_0;
	} else if (B <= 1900000.0) {
		tmp = -sqrt((2.0 * ((t_1 * F) * (2.0 * A)))) / t_1;
	} else {
		tmp = sqrt((F * (A - hypot(A, B)))) * -(sqrt(2.0) / B);
	}
	return tmp;
}

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) + Float64(Float64(A * C) * -4.0))
	t_1 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	tmp = 0.0
	if (B <= -7.2e+101)
		tmp = Float64(sqrt(fma(-4.0, Float64(C * F), Float64(F * Float64(4.0 * A)))) / B);
	elseif (B <= -5.2e-18)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_0 * Float64(F * Float64(A + Float64(C - hypot(B, Float64(A - C))))))))) / t_0);
	elseif (B <= 1900000.0)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(t_1 * F) * Float64(2.0 * A))))) / t_1);
	else
		tmp = Float64(sqrt(Float64(F * Float64(A - hypot(A, B)))) * Float64(-Float64(sqrt(2.0) / B)));
	end
	return tmp
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] + N[(N[(A * C), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -7.2e+101], N[(N[Sqrt[N[(-4.0 * N[(C * F), $MachinePrecision] + N[(F * N[(4.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision], If[LessEqual[B, -5.2e-18], N[((-N[Sqrt[N[(2.0 * N[(t$95$0 * N[(F * N[(A + N[(C - N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[B, 1900000.0], N[((-N[Sqrt[N[(2.0 * N[(N[(t$95$1 * F), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], N[(N[Sqrt[N[(F * N[(A - N[Sqrt[A ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision])), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < -7.20000000000000058e101

   1. Initial program 1.1%

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

   3. Simplified 1.1%

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

   4. Taylor expanded in A around -inf 0.4%

      \[\leadsto \frac{-\sqrt{\color{blue}{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right) + \left(4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right) + -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
   5. Step-by-step derivation

      1. fma-def 0.4%

         \[\leadsto \frac{-\sqrt{\color{blue}{\mathsf{fma}\left(-16, {A}^{2} \cdot \left(C \cdot F\right), 4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right) + -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      2. unpow2 0.4%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \color{blue}{\left(A \cdot A\right)} \cdot \left(C \cdot F\right), 4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right) + -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      3. fma-def 0.4%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \color{blue}{\mathsf{fma}\left(4, A \cdot \left(F \cdot {B}^{2}\right), -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      4. *-commutative 0.4%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \color{blue}{\left({B}^{2} \cdot F\right)}, -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      5. unpow2 0.4%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\color{blue}{\left(B \cdot B\right)} \cdot F\right), -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      6. *-commutative 0.4%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\left(B \cdot B\right) \cdot F\right), -4 \cdot \left(C \cdot \color{blue}{\left({B}^{2} \cdot F\right)}\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      7. unpow2 0.4%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\left(B \cdot B\right) \cdot F\right), -4 \cdot \left(C \cdot \left(\color{blue}{\left(B \cdot B\right)} \cdot F\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   6. Simplified 0.4%

      \[\leadsto \frac{-\sqrt{\color{blue}{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\left(B \cdot B\right) \cdot F\right), -4 \cdot \left(C \cdot \left(\left(B \cdot B\right) \cdot F\right)\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   7. Taylor expanded in B around -inf 6.6%

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

      1. associate-*r/ 6.6%

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

      2. *-rgt-identity 6.6%

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

      3. fma-def 6.6%

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

      4. *-commutative 6.6%

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

      5. associate-*r* 6.6%

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

   9. Simplified 6.6%

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

   if -7.20000000000000058e101 < B < -5.2000000000000001e-18

   1. Initial program 41.8%

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

   3. Simplified 41.8%

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

      1. distribute-frac-neg 41.8%

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

   5. Applied egg-rr 47.7%

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

   if -5.2000000000000001e-18 < B < 1.9e6

   1. Initial program 26.1%

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

   3. Simplified 26.0%

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

   4. Taylor expanded in A around -inf 30.7%

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

      1. *-commutative 30.7%

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

   6. Simplified 30.7%

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

   if 1.9e6 < B

   1. Initial program 18.3%

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

   3. Simplified 18.3%

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

   4. Taylor expanded in C around 0 23.6%

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

      1. mul-1-neg 23.6%

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

      2. *-commutative 23.6%

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

      3. +-commutative 23.6%

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

      4. unpow2 23.6%

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

      5. unpow2 23.6%

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

      6. hypot-def 54.8%

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

   6. Simplified 54.8%

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

4. Final simplification 33.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -7.2 \cdot 10^{+101}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, C \cdot F, F \cdot \left(4 \cdot A\right)\right)}}{B}\\ \mathbf{elif}\;B \leq -5.2 \cdot 10^{-18}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(B \cdot B + \left(A \cdot C\right) \cdot -4\right) \cdot \left(F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}{B \cdot B + \left(A \cdot C\right) \cdot -4}\\ \mathbf{elif}\;B \leq 1900000:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(2 \cdot A\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \end{array} \]

Alternative 4: 38.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B + \left(A \cdot C\right) \cdot -4\\ t_1 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -7.4 \cdot 10^{+101}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, C \cdot F, F \cdot \left(4 \cdot A\right)\right)}}{B}\\ \mathbf{elif}\;B \leq -7.6 \cdot 10^{-19}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 4200000:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(t_1 \cdot F\right) \cdot \left(2 \cdot A\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - B\right)}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (+ (* B B) (* (* A C) -4.0))) (t_1 (- (* B B) (* 4.0 (* A C)))))
   (if (<= B -7.4e+101)
     (/ (sqrt (fma -4.0 (* C F) (* F (* 4.0 A)))) B)
     (if (<= B -7.6e-19)
       (/ (- (sqrt (* 2.0 (* t_0 (* F (+ A (- C (hypot B (- A C))))))))) t_0)
       (if (<= B 4200000.0)
         (/ (- (sqrt (* 2.0 (* (* t_1 F) (* 2.0 A))))) t_1)
         (* (/ (sqrt 2.0) B) (- (sqrt (* F (- A B))))))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) + ((A * C) * -4.0);
	double t_1 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (B <= -7.4e+101) {
		tmp = sqrt(fma(-4.0, (C * F), (F * (4.0 * A)))) / B;
	} else if (B <= -7.6e-19) {
		tmp = -sqrt((2.0 * (t_0 * (F * (A + (C - hypot(B, (A - C)))))))) / t_0;
	} else if (B <= 4200000.0) {
		tmp = -sqrt((2.0 * ((t_1 * F) * (2.0 * A)))) / t_1;
	} else {
		tmp = (sqrt(2.0) / B) * -sqrt((F * (A - B)));
	}
	return tmp;
}

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) + Float64(Float64(A * C) * -4.0))
	t_1 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	tmp = 0.0
	if (B <= -7.4e+101)
		tmp = Float64(sqrt(fma(-4.0, Float64(C * F), Float64(F * Float64(4.0 * A)))) / B);
	elseif (B <= -7.6e-19)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_0 * Float64(F * Float64(A + Float64(C - hypot(B, Float64(A - C))))))))) / t_0);
	elseif (B <= 4200000.0)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(t_1 * F) * Float64(2.0 * A))))) / t_1);
	else
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(-sqrt(Float64(F * Float64(A - B)))));
	end
	return tmp
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] + N[(N[(A * C), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -7.4e+101], N[(N[Sqrt[N[(-4.0 * N[(C * F), $MachinePrecision] + N[(F * N[(4.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision], If[LessEqual[B, -7.6e-19], N[((-N[Sqrt[N[(2.0 * N[(t$95$0 * N[(F * N[(A + N[(C - N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[B, 4200000.0], N[((-N[Sqrt[N[(2.0 * N[(N[(t$95$1 * F), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * (-N[Sqrt[N[(F * N[(A - B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < -7.3999999999999995e101

   1. Initial program 1.1%

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

   3. Simplified 1.1%

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

   4. Taylor expanded in A around -inf 0.4%

      \[\leadsto \frac{-\sqrt{\color{blue}{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right) + \left(4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right) + -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
   5. Step-by-step derivation

      1. fma-def 0.4%

         \[\leadsto \frac{-\sqrt{\color{blue}{\mathsf{fma}\left(-16, {A}^{2} \cdot \left(C \cdot F\right), 4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right) + -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      2. unpow2 0.4%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \color{blue}{\left(A \cdot A\right)} \cdot \left(C \cdot F\right), 4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right) + -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      3. fma-def 0.4%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \color{blue}{\mathsf{fma}\left(4, A \cdot \left(F \cdot {B}^{2}\right), -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      4. *-commutative 0.4%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \color{blue}{\left({B}^{2} \cdot F\right)}, -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      5. unpow2 0.4%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\color{blue}{\left(B \cdot B\right)} \cdot F\right), -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      6. *-commutative 0.4%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\left(B \cdot B\right) \cdot F\right), -4 \cdot \left(C \cdot \color{blue}{\left({B}^{2} \cdot F\right)}\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      7. unpow2 0.4%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\left(B \cdot B\right) \cdot F\right), -4 \cdot \left(C \cdot \left(\color{blue}{\left(B \cdot B\right)} \cdot F\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   6. Simplified 0.4%

      \[\leadsto \frac{-\sqrt{\color{blue}{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\left(B \cdot B\right) \cdot F\right), -4 \cdot \left(C \cdot \left(\left(B \cdot B\right) \cdot F\right)\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   7. Taylor expanded in B around -inf 6.6%

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

      1. associate-*r/ 6.6%

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

      2. *-rgt-identity 6.6%

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

      3. fma-def 6.6%

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

      4. *-commutative 6.6%

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

      5. associate-*r* 6.6%

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

   9. Simplified 6.6%

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

   if -7.3999999999999995e101 < B < -7.6e-19

   1. Initial program 41.8%

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

   3. Simplified 41.8%

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

      1. distribute-frac-neg 41.8%

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

   5. Applied egg-rr 47.7%

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

   if -7.6e-19 < B < 4.2e6

   1. Initial program 26.1%

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

   3. Simplified 26.0%

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

   4. Taylor expanded in A around -inf 30.7%

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

      1. *-commutative 30.7%

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

   6. Simplified 30.7%

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

   if 4.2e6 < B

   1. Initial program 18.3%

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

   3. Simplified 18.3%

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

   4. Taylor expanded in B around inf 17.5%

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

   5. Taylor expanded in A around 0 17.1%

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

      1. unpow2 17.1%

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

      2. associate-*r* 17.1%

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

   7. Simplified 17.1%

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

   8. Taylor expanded in C around 0 52.8%

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

      1. mul-1-neg 52.8%

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

      2. *-commutative 52.8%

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

   10. Simplified 52.8%

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

4. Final simplification 32.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -7.4 \cdot 10^{+101}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, C \cdot F, F \cdot \left(4 \cdot A\right)\right)}}{B}\\ \mathbf{elif}\;B \leq -7.6 \cdot 10^{-19}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(B \cdot B + \left(A \cdot C\right) \cdot -4\right) \cdot \left(F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}{B \cdot B + \left(A \cdot C\right) \cdot -4}\\ \mathbf{elif}\;B \leq 4200000:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(2 \cdot A\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - B\right)}\right)\\ \end{array} \]

Alternative 5: 38.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ t_1 := t_0 \cdot F\\ \mathbf{if}\;B \leq -7.5 \cdot 10^{+110}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, C \cdot F, F \cdot \left(4 \cdot A\right)\right)}}{B}\\ \mathbf{elif}\;B \leq -2.6 \cdot 10^{-18}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 2300000:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(2 \cdot A\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - B\right)}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* A C)))) (t_1 (* t_0 F)))
   (if (<= B -7.5e+110)
     (/ (sqrt (fma -4.0 (* C F) (* F (* 4.0 A)))) B)
     (if (<= B -2.6e-18)
       (/ (- (sqrt (* 2.0 (* t_1 (- A (hypot A B)))))) t_0)
       (if (<= B 2300000.0)
         (/ (- (sqrt (* 2.0 (* t_1 (* 2.0 A))))) t_0)
         (* (/ (sqrt 2.0) B) (- (sqrt (* F (- A B))))))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double t_1 = t_0 * F;
	double tmp;
	if (B <= -7.5e+110) {
		tmp = sqrt(fma(-4.0, (C * F), (F * (4.0 * A)))) / B;
	} else if (B <= -2.6e-18) {
		tmp = -sqrt((2.0 * (t_1 * (A - hypot(A, B))))) / t_0;
	} else if (B <= 2300000.0) {
		tmp = -sqrt((2.0 * (t_1 * (2.0 * A)))) / t_0;
	} else {
		tmp = (sqrt(2.0) / B) * -sqrt((F * (A - B)));
	}
	return tmp;
}

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	t_1 = Float64(t_0 * F)
	tmp = 0.0
	if (B <= -7.5e+110)
		tmp = Float64(sqrt(fma(-4.0, Float64(C * F), Float64(F * Float64(4.0 * A)))) / B);
	elseif (B <= -2.6e-18)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_1 * Float64(A - hypot(A, B)))))) / t_0);
	elseif (B <= 2300000.0)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_1 * Float64(2.0 * A))))) / t_0);
	else
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(-sqrt(Float64(F * Float64(A - B)))));
	end
	return tmp
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * F), $MachinePrecision]}, If[LessEqual[B, -7.5e+110], N[(N[Sqrt[N[(-4.0 * N[(C * F), $MachinePrecision] + N[(F * N[(4.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision], If[LessEqual[B, -2.6e-18], N[((-N[Sqrt[N[(2.0 * N[(t$95$1 * N[(A - N[Sqrt[A ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[B, 2300000.0], N[((-N[Sqrt[N[(2.0 * N[(t$95$1 * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * (-N[Sqrt[N[(F * N[(A - B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < -7.5e110

   1. Initial program 0.9%

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

   3. Simplified 1.0%

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

   4. Taylor expanded in A around -inf 0.4%

      \[\leadsto \frac{-\sqrt{\color{blue}{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right) + \left(4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right) + -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
   5. Step-by-step derivation

      1. fma-def 0.4%

         \[\leadsto \frac{-\sqrt{\color{blue}{\mathsf{fma}\left(-16, {A}^{2} \cdot \left(C \cdot F\right), 4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right) + -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      2. unpow2 0.4%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \color{blue}{\left(A \cdot A\right)} \cdot \left(C \cdot F\right), 4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right) + -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      3. fma-def 0.4%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \color{blue}{\mathsf{fma}\left(4, A \cdot \left(F \cdot {B}^{2}\right), -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      4. *-commutative 0.4%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \color{blue}{\left({B}^{2} \cdot F\right)}, -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      5. unpow2 0.4%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\color{blue}{\left(B \cdot B\right)} \cdot F\right), -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      6. *-commutative 0.4%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\left(B \cdot B\right) \cdot F\right), -4 \cdot \left(C \cdot \color{blue}{\left({B}^{2} \cdot F\right)}\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      7. unpow2 0.4%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\left(B \cdot B\right) \cdot F\right), -4 \cdot \left(C \cdot \left(\color{blue}{\left(B \cdot B\right)} \cdot F\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   6. Simplified 0.4%

      \[\leadsto \frac{-\sqrt{\color{blue}{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\left(B \cdot B\right) \cdot F\right), -4 \cdot \left(C \cdot \left(\left(B \cdot B\right) \cdot F\right)\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   7. Taylor expanded in B around -inf 6.9%

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

      1. associate-*r/ 6.9%

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

      2. *-rgt-identity 6.9%

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

      3. fma-def 6.9%

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

      4. *-commutative 6.9%

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

      5. associate-*r* 6.9%

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

   9. Simplified 6.9%

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

   if -7.5e110 < B < -2.6e-18

   1. Initial program 38.4%

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

   3. Simplified 38.4%

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

   4. Taylor expanded in C around 0 35.0%

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

      1. +-commutative 35.0%

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

      2. unpow2 35.0%

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

      3. unpow2 35.0%

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

      4. hypot-def 39.4%

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

   6. Simplified 39.4%

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

   if -2.6e-18 < B < 2.3e6

   1. Initial program 26.1%

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

   3. Simplified 26.0%

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

   4. Taylor expanded in A around -inf 30.7%

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

      1. *-commutative 30.7%

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

   6. Simplified 30.7%

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

   if 2.3e6 < B

   1. Initial program 18.3%

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

   3. Simplified 18.3%

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

   4. Taylor expanded in B around inf 17.5%

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

   5. Taylor expanded in A around 0 17.1%

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

      1. unpow2 17.1%

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

      2. associate-*r* 17.1%

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

   7. Simplified 17.1%

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

   8. Taylor expanded in C around 0 52.8%

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

      1. mul-1-neg 52.8%

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

      2. *-commutative 52.8%

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

   10. Simplified 52.8%

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

4. Final simplification 32.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -7.5 \cdot 10^{+110}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, C \cdot F, F \cdot \left(4 \cdot A\right)\right)}}{B}\\ \mathbf{elif}\;B \leq -2.6 \cdot 10^{-18}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 2300000:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(2 \cdot A\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - B\right)}\right)\\ \end{array} \]

Alternative 6: 37.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ t_1 := t_0 \cdot F\\ \mathbf{if}\;B \leq -3.05 \cdot 10^{+100}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, C \cdot F, F \cdot \left(4 \cdot A\right)\right)}}{B}\\ \mathbf{elif}\;B \leq -2.9 \cdot 10^{-18}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(B + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq -1.35 \cdot 10^{-114}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(2 \cdot A\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 1400000:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(2 \cdot A\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - B\right)}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* A C)))) (t_1 (* t_0 F)))
   (if (<= B -3.05e+100)
     (/ (sqrt (fma -4.0 (* C F) (* F (* 4.0 A)))) B)
     (if (<= B -2.9e-18)
       (/ (- (sqrt (* 2.0 (* t_1 (+ A (+ B C)))))) t_0)
       (if (<= B -1.35e-114)
         (/
          (- (sqrt (* 2.0 (* F (* (fma B B (* C (* A -4.0))) (* 2.0 A))))))
          t_0)
         (if (<= B 1400000.0)
           (/ (- (sqrt (* 2.0 (* t_1 (* 2.0 A))))) t_0)
           (* (/ (sqrt 2.0) B) (- (sqrt (* F (- A B)))))))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double t_1 = t_0 * F;
	double tmp;
	if (B <= -3.05e+100) {
		tmp = sqrt(fma(-4.0, (C * F), (F * (4.0 * A)))) / B;
	} else if (B <= -2.9e-18) {
		tmp = -sqrt((2.0 * (t_1 * (A + (B + C))))) / t_0;
	} else if (B <= -1.35e-114) {
		tmp = -sqrt((2.0 * (F * (fma(B, B, (C * (A * -4.0))) * (2.0 * A))))) / t_0;
	} else if (B <= 1400000.0) {
		tmp = -sqrt((2.0 * (t_1 * (2.0 * A)))) / t_0;
	} else {
		tmp = (sqrt(2.0) / B) * -sqrt((F * (A - B)));
	}
	return tmp;
}

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	t_1 = Float64(t_0 * F)
	tmp = 0.0
	if (B <= -3.05e+100)
		tmp = Float64(sqrt(fma(-4.0, Float64(C * F), Float64(F * Float64(4.0 * A)))) / B);
	elseif (B <= -2.9e-18)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_1 * Float64(A + Float64(B + C)))))) / t_0);
	elseif (B <= -1.35e-114)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(F * Float64(fma(B, B, Float64(C * Float64(A * -4.0))) * Float64(2.0 * A)))))) / t_0);
	elseif (B <= 1400000.0)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_1 * Float64(2.0 * A))))) / t_0);
	else
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(-sqrt(Float64(F * Float64(A - B)))));
	end
	return tmp
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * F), $MachinePrecision]}, If[LessEqual[B, -3.05e+100], N[(N[Sqrt[N[(-4.0 * N[(C * F), $MachinePrecision] + N[(F * N[(4.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision], If[LessEqual[B, -2.9e-18], N[((-N[Sqrt[N[(2.0 * N[(t$95$1 * N[(A + N[(B + C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[B, -1.35e-114], N[((-N[Sqrt[N[(2.0 * N[(F * N[(N[(B * B + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[B, 1400000.0], N[((-N[Sqrt[N[(2.0 * N[(t$95$1 * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * (-N[Sqrt[N[(F * N[(A - B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if B < -3.05e100

   1. Initial program 1.1%

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

   3. Simplified 1.1%

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

   4. Taylor expanded in A around -inf 0.4%

      \[\leadsto \frac{-\sqrt{\color{blue}{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right) + \left(4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right) + -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
   5. Step-by-step derivation

      1. fma-def 0.4%

         \[\leadsto \frac{-\sqrt{\color{blue}{\mathsf{fma}\left(-16, {A}^{2} \cdot \left(C \cdot F\right), 4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right) + -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      2. unpow2 0.4%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \color{blue}{\left(A \cdot A\right)} \cdot \left(C \cdot F\right), 4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right) + -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      3. fma-def 0.4%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \color{blue}{\mathsf{fma}\left(4, A \cdot \left(F \cdot {B}^{2}\right), -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      4. *-commutative 0.4%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \color{blue}{\left({B}^{2} \cdot F\right)}, -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      5. unpow2 0.4%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\color{blue}{\left(B \cdot B\right)} \cdot F\right), -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      6. *-commutative 0.4%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\left(B \cdot B\right) \cdot F\right), -4 \cdot \left(C \cdot \color{blue}{\left({B}^{2} \cdot F\right)}\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      7. unpow2 0.4%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\left(B \cdot B\right) \cdot F\right), -4 \cdot \left(C \cdot \left(\color{blue}{\left(B \cdot B\right)} \cdot F\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   6. Simplified 0.4%

      \[\leadsto \frac{-\sqrt{\color{blue}{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\left(B \cdot B\right) \cdot F\right), -4 \cdot \left(C \cdot \left(\left(B \cdot B\right) \cdot F\right)\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   7. Taylor expanded in B around -inf 6.6%

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

      1. associate-*r/ 6.6%

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

      2. *-rgt-identity 6.6%

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

      3. fma-def 6.6%

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

      4. *-commutative 6.6%

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

      5. associate-*r* 6.6%

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

   9. Simplified 6.6%

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

   if -3.05e100 < B < -2.9e-18

   1. Initial program 41.8%

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

   3. Simplified 41.8%

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

   4. Taylor expanded in B around -inf 38.2%

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

   if -2.9e-18 < B < -1.35e-114

   1. Initial program 18.3%

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

   3. Simplified 18.3%

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

   4. Taylor expanded in A around -inf 29.5%

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

      1. *-commutative 29.5%

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

   6. Simplified 29.5%

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

      1. sqrt-prod 29.5%

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

      2. associate-*l* 24.2%

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

      3. cancel-sign-sub-inv 24.2%

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

      4. metadata-eval 24.2%

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

      5. *-commutative 24.2%

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

   8. Applied egg-rr 24.2%

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

      1. pow1 24.2%

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

      2. sqrt-unprod 24.2%

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

      3. metadata-eval 24.2%

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

      4. cancel-sign-sub-inv 24.2%

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

      5. *-commutative 24.2%

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

      6. cancel-sign-sub-inv 24.2%

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

      7. metadata-eval 24.2%

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

      8. fma-def 24.2%

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

   10. Applied egg-rr 24.2%

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

       1. unpow1 24.2%

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

       2. associate-*l* 29.5%

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

       3. associate-*r* 29.5%

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

   12. Simplified 29.5%

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

   if -1.35e-114 < B < 1.4e6

   1. Initial program 27.3%

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

   3. Simplified 27.3%

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

   4. Taylor expanded in A around -inf 30.9%

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

      1. *-commutative 30.9%

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

   6. Simplified 30.9%

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

   if 1.4e6 < B

   1. Initial program 18.3%

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

   3. Simplified 18.3%

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

   4. Taylor expanded in B around inf 17.5%

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

   5. Taylor expanded in A around 0 17.1%

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

      1. unpow2 17.1%

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

      2. associate-*r* 17.1%

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

   7. Simplified 17.1%

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

   8. Taylor expanded in C around 0 52.8%

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

      1. mul-1-neg 52.8%

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

      2. *-commutative 52.8%

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

   10. Simplified 52.8%

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

4. Final simplification 32.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -3.05 \cdot 10^{+100}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, C \cdot F, F \cdot \left(4 \cdot A\right)\right)}}{B}\\ \mathbf{elif}\;B \leq -2.9 \cdot 10^{-18}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(A + \left(B + C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq -1.35 \cdot 10^{-114}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(2 \cdot A\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 1400000:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(2 \cdot A\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - B\right)}\right)\\ \end{array} \]

Alternative 7: 36.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -2.3 \cdot 10^{+55}:\\ \;\;\;\;\sqrt{-4 \cdot \left(C \cdot F\right) + 4 \cdot \left(A \cdot F\right)} \cdot \frac{1}{B}\\ \mathbf{elif}\;B \leq 4400000:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(t_0 \cdot F\right) \cdot \left(2 \cdot A\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - B\right)}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* A C)))))
   (if (<= B -2.3e+55)
     (* (sqrt (+ (* -4.0 (* C F)) (* 4.0 (* A F)))) (/ 1.0 B))
     (if (<= B 4400000.0)
       (/ (- (sqrt (* 2.0 (* (* t_0 F) (* 2.0 A))))) t_0)
       (* (/ (sqrt 2.0) B) (- (sqrt (* F (- A B)))))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (B <= -2.3e+55) {
		tmp = sqrt(((-4.0 * (C * F)) + (4.0 * (A * F)))) * (1.0 / B);
	} else if (B <= 4400000.0) {
		tmp = -sqrt((2.0 * ((t_0 * F) * (2.0 * A)))) / t_0;
	} else {
		tmp = (sqrt(2.0) / B) * -sqrt((F * (A - B)));
	}
	return tmp;
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b * b) - (4.0d0 * (a * c))
    if (b <= (-2.3d+55)) then
        tmp = sqrt((((-4.0d0) * (c * f)) + (4.0d0 * (a * f)))) * (1.0d0 / b)
    else if (b <= 4400000.0d0) then
        tmp = -sqrt((2.0d0 * ((t_0 * f) * (2.0d0 * a)))) / t_0
    else
        tmp = (sqrt(2.0d0) / b) * -sqrt((f * (a - b)))
    end if
    code = tmp
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (B <= -2.3e+55) {
		tmp = Math.sqrt(((-4.0 * (C * F)) + (4.0 * (A * F)))) * (1.0 / B);
	} else if (B <= 4400000.0) {
		tmp = -Math.sqrt((2.0 * ((t_0 * F) * (2.0 * A)))) / t_0;
	} else {
		tmp = (Math.sqrt(2.0) / B) * -Math.sqrt((F * (A - B)));
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (A * C))
	tmp = 0
	if B <= -2.3e+55:
		tmp = math.sqrt(((-4.0 * (C * F)) + (4.0 * (A * F)))) * (1.0 / B)
	elif B <= 4400000.0:
		tmp = -math.sqrt((2.0 * ((t_0 * F) * (2.0 * A)))) / t_0
	else:
		tmp = (math.sqrt(2.0) / B) * -math.sqrt((F * (A - B)))
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	tmp = 0.0
	if (B <= -2.3e+55)
		tmp = Float64(sqrt(Float64(Float64(-4.0 * Float64(C * F)) + Float64(4.0 * Float64(A * F)))) * Float64(1.0 / B));
	elseif (B <= 4400000.0)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(t_0 * F) * Float64(2.0 * A))))) / t_0);
	else
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(-sqrt(Float64(F * Float64(A - B)))));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (A * C));
	tmp = 0.0;
	if (B <= -2.3e+55)
		tmp = sqrt(((-4.0 * (C * F)) + (4.0 * (A * F)))) * (1.0 / B);
	elseif (B <= 4400000.0)
		tmp = -sqrt((2.0 * ((t_0 * F) * (2.0 * A)))) / t_0;
	else
		tmp = (sqrt(2.0) / B) * -sqrt((F * (A - B)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -2.3e+55], N[(N[Sqrt[N[(N[(-4.0 * N[(C * F), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(A * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 4400000.0], N[((-N[Sqrt[N[(2.0 * N[(N[(t$95$0 * F), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * (-N[Sqrt[N[(F * N[(A - B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < -2.29999999999999987e55

   1. Initial program 6.7%

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

   3. Simplified 6.6%

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

   4. Taylor expanded in A around -inf 0.7%

      \[\leadsto \frac{-\sqrt{\color{blue}{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right) + \left(4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right) + -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
   5. Step-by-step derivation

      1. fma-def 0.7%

         \[\leadsto \frac{-\sqrt{\color{blue}{\mathsf{fma}\left(-16, {A}^{2} \cdot \left(C \cdot F\right), 4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right) + -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      2. unpow2 0.7%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \color{blue}{\left(A \cdot A\right)} \cdot \left(C \cdot F\right), 4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right) + -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      3. fma-def 0.7%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \color{blue}{\mathsf{fma}\left(4, A \cdot \left(F \cdot {B}^{2}\right), -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      4. *-commutative 0.7%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \color{blue}{\left({B}^{2} \cdot F\right)}, -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      5. unpow2 0.7%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\color{blue}{\left(B \cdot B\right)} \cdot F\right), -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      6. *-commutative 0.7%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\left(B \cdot B\right) \cdot F\right), -4 \cdot \left(C \cdot \color{blue}{\left({B}^{2} \cdot F\right)}\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      7. unpow2 0.7%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\left(B \cdot B\right) \cdot F\right), -4 \cdot \left(C \cdot \left(\color{blue}{\left(B \cdot B\right)} \cdot F\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   6. Simplified 0.7%

      \[\leadsto \frac{-\sqrt{\color{blue}{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\left(B \cdot B\right) \cdot F\right), -4 \cdot \left(C \cdot \left(\left(B \cdot B\right) \cdot F\right)\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   7. Taylor expanded in B around -inf 6.2%

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

   if -2.29999999999999987e55 < B < 4.4e6

   1. Initial program 27.2%

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

   3. Simplified 27.2%

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

   4. Taylor expanded in A around -inf 29.4%

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

      1. *-commutative 29.4%

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

   6. Simplified 29.4%

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

   if 4.4e6 < B

   1. Initial program 18.3%

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

   3. Simplified 18.3%

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

   4. Taylor expanded in B around inf 17.5%

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

   5. Taylor expanded in A around 0 17.1%

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

      1. unpow2 17.1%

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

      2. associate-*r* 17.1%

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

   7. Simplified 17.1%

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

   8. Taylor expanded in C around 0 52.8%

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

      1. mul-1-neg 52.8%

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

      2. *-commutative 52.8%

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

   10. Simplified 52.8%

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

4. Final simplification 30.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -2.3 \cdot 10^{+55}:\\ \;\;\;\;\sqrt{-4 \cdot \left(C \cdot F\right) + 4 \cdot \left(A \cdot F\right)} \cdot \frac{1}{B}\\ \mathbf{elif}\;B \leq 4400000:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(2 \cdot A\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - B\right)}\right)\\ \end{array} \]

Alternative 8: 36.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -3.1 \cdot 10^{+55}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, C \cdot F, F \cdot \left(4 \cdot A\right)\right)}}{B}\\ \mathbf{elif}\;B \leq 2020000:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(t_0 \cdot F\right) \cdot \left(2 \cdot A\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - B\right)}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* A C)))))
   (if (<= B -3.1e+55)
     (/ (sqrt (fma -4.0 (* C F) (* F (* 4.0 A)))) B)
     (if (<= B 2020000.0)
       (/ (- (sqrt (* 2.0 (* (* t_0 F) (* 2.0 A))))) t_0)
       (* (/ (sqrt 2.0) B) (- (sqrt (* F (- A B)))))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (B <= -3.1e+55) {
		tmp = sqrt(fma(-4.0, (C * F), (F * (4.0 * A)))) / B;
	} else if (B <= 2020000.0) {
		tmp = -sqrt((2.0 * ((t_0 * F) * (2.0 * A)))) / t_0;
	} else {
		tmp = (sqrt(2.0) / B) * -sqrt((F * (A - B)));
	}
	return tmp;
}

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	tmp = 0.0
	if (B <= -3.1e+55)
		tmp = Float64(sqrt(fma(-4.0, Float64(C * F), Float64(F * Float64(4.0 * A)))) / B);
	elseif (B <= 2020000.0)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(t_0 * F) * Float64(2.0 * A))))) / t_0);
	else
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(-sqrt(Float64(F * Float64(A - B)))));
	end
	return tmp
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -3.1e+55], N[(N[Sqrt[N[(-4.0 * N[(C * F), $MachinePrecision] + N[(F * N[(4.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision], If[LessEqual[B, 2020000.0], N[((-N[Sqrt[N[(2.0 * N[(N[(t$95$0 * F), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * (-N[Sqrt[N[(F * N[(A - B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < -3.09999999999999994e55

   1. Initial program 6.7%

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

   3. Simplified 6.6%

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

   4. Taylor expanded in A around -inf 0.7%

      \[\leadsto \frac{-\sqrt{\color{blue}{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right) + \left(4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right) + -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
   5. Step-by-step derivation

      1. fma-def 0.7%

         \[\leadsto \frac{-\sqrt{\color{blue}{\mathsf{fma}\left(-16, {A}^{2} \cdot \left(C \cdot F\right), 4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right) + -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      2. unpow2 0.7%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \color{blue}{\left(A \cdot A\right)} \cdot \left(C \cdot F\right), 4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right) + -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      3. fma-def 0.7%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \color{blue}{\mathsf{fma}\left(4, A \cdot \left(F \cdot {B}^{2}\right), -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      4. *-commutative 0.7%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \color{blue}{\left({B}^{2} \cdot F\right)}, -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      5. unpow2 0.7%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\color{blue}{\left(B \cdot B\right)} \cdot F\right), -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      6. *-commutative 0.7%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\left(B \cdot B\right) \cdot F\right), -4 \cdot \left(C \cdot \color{blue}{\left({B}^{2} \cdot F\right)}\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      7. unpow2 0.7%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\left(B \cdot B\right) \cdot F\right), -4 \cdot \left(C \cdot \left(\color{blue}{\left(B \cdot B\right)} \cdot F\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   6. Simplified 0.7%

      \[\leadsto \frac{-\sqrt{\color{blue}{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\left(B \cdot B\right) \cdot F\right), -4 \cdot \left(C \cdot \left(\left(B \cdot B\right) \cdot F\right)\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   7. Taylor expanded in B around -inf 6.2%

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

      1. associate-*r/ 6.2%

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

      2. *-rgt-identity 6.2%

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

      3. fma-def 6.2%

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

      4. *-commutative 6.2%

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

      5. associate-*r* 6.2%

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

   9. Simplified 6.2%

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

   if -3.09999999999999994e55 < B < 2.02e6

   1. Initial program 27.2%

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

   3. Simplified 27.2%

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

   4. Taylor expanded in A around -inf 29.4%

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

      1. *-commutative 29.4%

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

   6. Simplified 29.4%

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

   if 2.02e6 < B

   1. Initial program 18.3%

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

   3. Simplified 18.3%

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

   4. Taylor expanded in B around inf 17.5%

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

   5. Taylor expanded in A around 0 17.1%

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

      1. unpow2 17.1%

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

      2. associate-*r* 17.1%

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

   7. Simplified 17.1%

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

   8. Taylor expanded in C around 0 52.8%

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

      1. mul-1-neg 52.8%

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

      2. *-commutative 52.8%

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

   10. Simplified 52.8%

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

4. Final simplification 30.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -3.1 \cdot 10^{+55}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, C \cdot F, F \cdot \left(4 \cdot A\right)\right)}}{B}\\ \mathbf{elif}\;B \leq 2020000:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(2 \cdot A\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - B\right)}\right)\\ \end{array} \]

Alternative 9: 29.1% accurate, 4.5× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ t_1 := t_0 \cdot F\\ \mathbf{if}\;B \leq -1 \cdot 10^{+54}:\\ \;\;\;\;\sqrt{-4 \cdot \left(C \cdot F\right) + 4 \cdot \left(A \cdot F\right)} \cdot \frac{1}{B}\\ \mathbf{elif}\;B \leq 25:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(2 \cdot A\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(\left(A + C\right) - \left(B + 0.5 \cdot \frac{C \cdot C}{B}\right)\right)\right)}}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* A C)))) (t_1 (* t_0 F)))
   (if (<= B -1e+54)
     (* (sqrt (+ (* -4.0 (* C F)) (* 4.0 (* A F)))) (/ 1.0 B))
     (if (<= B 25.0)
       (/ (- (sqrt (* 2.0 (* t_1 (* 2.0 A))))) t_0)
       (/
        (- (sqrt (* 2.0 (* t_1 (- (+ A C) (+ B (* 0.5 (/ (* C C) B))))))))
        t_0)))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double t_1 = t_0 * F;
	double tmp;
	if (B <= -1e+54) {
		tmp = sqrt(((-4.0 * (C * F)) + (4.0 * (A * F)))) * (1.0 / B);
	} else if (B <= 25.0) {
		tmp = -sqrt((2.0 * (t_1 * (2.0 * A)))) / t_0;
	} else {
		tmp = -sqrt((2.0 * (t_1 * ((A + C) - (B + (0.5 * ((C * C) / B))))))) / t_0;
	}
	return tmp;
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (b * b) - (4.0d0 * (a * c))
    t_1 = t_0 * f
    if (b <= (-1d+54)) then
        tmp = sqrt((((-4.0d0) * (c * f)) + (4.0d0 * (a * f)))) * (1.0d0 / b)
    else if (b <= 25.0d0) then
        tmp = -sqrt((2.0d0 * (t_1 * (2.0d0 * a)))) / t_0
    else
        tmp = -sqrt((2.0d0 * (t_1 * ((a + c) - (b + (0.5d0 * ((c * c) / b))))))) / t_0
    end if
    code = tmp
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double t_1 = t_0 * F;
	double tmp;
	if (B <= -1e+54) {
		tmp = Math.sqrt(((-4.0 * (C * F)) + (4.0 * (A * F)))) * (1.0 / B);
	} else if (B <= 25.0) {
		tmp = -Math.sqrt((2.0 * (t_1 * (2.0 * A)))) / t_0;
	} else {
		tmp = -Math.sqrt((2.0 * (t_1 * ((A + C) - (B + (0.5 * ((C * C) / B))))))) / t_0;
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (A * C))
	t_1 = t_0 * F
	tmp = 0
	if B <= -1e+54:
		tmp = math.sqrt(((-4.0 * (C * F)) + (4.0 * (A * F)))) * (1.0 / B)
	elif B <= 25.0:
		tmp = -math.sqrt((2.0 * (t_1 * (2.0 * A)))) / t_0
	else:
		tmp = -math.sqrt((2.0 * (t_1 * ((A + C) - (B + (0.5 * ((C * C) / B))))))) / t_0
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	t_1 = Float64(t_0 * F)
	tmp = 0.0
	if (B <= -1e+54)
		tmp = Float64(sqrt(Float64(Float64(-4.0 * Float64(C * F)) + Float64(4.0 * Float64(A * F)))) * Float64(1.0 / B));
	elseif (B <= 25.0)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_1 * Float64(2.0 * A))))) / t_0);
	else
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_1 * Float64(Float64(A + C) - Float64(B + Float64(0.5 * Float64(Float64(C * C) / B)))))))) / t_0);
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (A * C));
	t_1 = t_0 * F;
	tmp = 0.0;
	if (B <= -1e+54)
		tmp = sqrt(((-4.0 * (C * F)) + (4.0 * (A * F)))) * (1.0 / B);
	elseif (B <= 25.0)
		tmp = -sqrt((2.0 * (t_1 * (2.0 * A)))) / t_0;
	else
		tmp = -sqrt((2.0 * (t_1 * ((A + C) - (B + (0.5 * ((C * C) / B))))))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * F), $MachinePrecision]}, If[LessEqual[B, -1e+54], N[(N[Sqrt[N[(N[(-4.0 * N[(C * F), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(A * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 25.0], N[((-N[Sqrt[N[(2.0 * N[(t$95$1 * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[((-N[Sqrt[N[(2.0 * N[(t$95$1 * N[(N[(A + C), $MachinePrecision] - N[(B + N[(0.5 * N[(N[(C * C), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if B < -1.0000000000000001e54

   1. Initial program 6.7%

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

   3. Simplified 6.6%

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

   4. Taylor expanded in A around -inf 0.7%

      \[\leadsto \frac{-\sqrt{\color{blue}{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right) + \left(4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right) + -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
   5. Step-by-step derivation

      1. fma-def 0.7%

         \[\leadsto \frac{-\sqrt{\color{blue}{\mathsf{fma}\left(-16, {A}^{2} \cdot \left(C \cdot F\right), 4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right) + -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      2. unpow2 0.7%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \color{blue}{\left(A \cdot A\right)} \cdot \left(C \cdot F\right), 4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right) + -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      3. fma-def 0.7%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \color{blue}{\mathsf{fma}\left(4, A \cdot \left(F \cdot {B}^{2}\right), -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      4. *-commutative 0.7%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \color{blue}{\left({B}^{2} \cdot F\right)}, -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      5. unpow2 0.7%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\color{blue}{\left(B \cdot B\right)} \cdot F\right), -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      6. *-commutative 0.7%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\left(B \cdot B\right) \cdot F\right), -4 \cdot \left(C \cdot \color{blue}{\left({B}^{2} \cdot F\right)}\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      7. unpow2 0.7%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\left(B \cdot B\right) \cdot F\right), -4 \cdot \left(C \cdot \left(\color{blue}{\left(B \cdot B\right)} \cdot F\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   6. Simplified 0.7%

      \[\leadsto \frac{-\sqrt{\color{blue}{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\left(B \cdot B\right) \cdot F\right), -4 \cdot \left(C \cdot \left(\left(B \cdot B\right) \cdot F\right)\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   7. Taylor expanded in B around -inf 6.2%

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

   if -1.0000000000000001e54 < B < 25

   1. Initial program 26.2%

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

   3. Simplified 26.1%

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

   4. Taylor expanded in A around -inf 29.1%

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

      1. *-commutative 29.1%

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

   6. Simplified 29.1%

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

   if 25 < B

   1. Initial program 20.8%

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

   3. Simplified 20.8%

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

   4. Taylor expanded in B around inf 18.6%

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

   5. Taylor expanded in A around 0 18.2%

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

      1. unpow2 18.2%

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

   7. Simplified 18.2%

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

4. Final simplification 21.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -1 \cdot 10^{+54}:\\ \;\;\;\;\sqrt{-4 \cdot \left(C \cdot F\right) + 4 \cdot \left(A \cdot F\right)} \cdot \frac{1}{B}\\ \mathbf{elif}\;B \leq 25:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(2 \cdot A\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(\left(A + C\right) - \left(B + 0.5 \cdot \frac{C \cdot C}{B}\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \end{array} \]

Alternative 10: 29.0% accurate, 4.7× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ t_1 := t_0 \cdot F\\ \mathbf{if}\;B \leq -1.06 \cdot 10^{+54}:\\ \;\;\;\;\sqrt{-4 \cdot \left(C \cdot F\right) + 4 \cdot \left(A \cdot F\right)} \cdot \frac{1}{B}\\ \mathbf{elif}\;B \leq 18:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(2 \cdot A\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(\left(A + C\right) - B\right)\right)}}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* A C)))) (t_1 (* t_0 F)))
   (if (<= B -1.06e+54)
     (* (sqrt (+ (* -4.0 (* C F)) (* 4.0 (* A F)))) (/ 1.0 B))
     (if (<= B 18.0)
       (/ (- (sqrt (* 2.0 (* t_1 (* 2.0 A))))) t_0)
       (/ (- (sqrt (* 2.0 (* t_1 (- (+ A C) B))))) t_0)))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double t_1 = t_0 * F;
	double tmp;
	if (B <= -1.06e+54) {
		tmp = sqrt(((-4.0 * (C * F)) + (4.0 * (A * F)))) * (1.0 / B);
	} else if (B <= 18.0) {
		tmp = -sqrt((2.0 * (t_1 * (2.0 * A)))) / t_0;
	} else {
		tmp = -sqrt((2.0 * (t_1 * ((A + C) - B)))) / t_0;
	}
	return tmp;
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (b * b) - (4.0d0 * (a * c))
    t_1 = t_0 * f
    if (b <= (-1.06d+54)) then
        tmp = sqrt((((-4.0d0) * (c * f)) + (4.0d0 * (a * f)))) * (1.0d0 / b)
    else if (b <= 18.0d0) then
        tmp = -sqrt((2.0d0 * (t_1 * (2.0d0 * a)))) / t_0
    else
        tmp = -sqrt((2.0d0 * (t_1 * ((a + c) - b)))) / t_0
    end if
    code = tmp
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double t_1 = t_0 * F;
	double tmp;
	if (B <= -1.06e+54) {
		tmp = Math.sqrt(((-4.0 * (C * F)) + (4.0 * (A * F)))) * (1.0 / B);
	} else if (B <= 18.0) {
		tmp = -Math.sqrt((2.0 * (t_1 * (2.0 * A)))) / t_0;
	} else {
		tmp = -Math.sqrt((2.0 * (t_1 * ((A + C) - B)))) / t_0;
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (A * C))
	t_1 = t_0 * F
	tmp = 0
	if B <= -1.06e+54:
		tmp = math.sqrt(((-4.0 * (C * F)) + (4.0 * (A * F)))) * (1.0 / B)
	elif B <= 18.0:
		tmp = -math.sqrt((2.0 * (t_1 * (2.0 * A)))) / t_0
	else:
		tmp = -math.sqrt((2.0 * (t_1 * ((A + C) - B)))) / t_0
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	t_1 = Float64(t_0 * F)
	tmp = 0.0
	if (B <= -1.06e+54)
		tmp = Float64(sqrt(Float64(Float64(-4.0 * Float64(C * F)) + Float64(4.0 * Float64(A * F)))) * Float64(1.0 / B));
	elseif (B <= 18.0)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_1 * Float64(2.0 * A))))) / t_0);
	else
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_1 * Float64(Float64(A + C) - B))))) / t_0);
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (A * C));
	t_1 = t_0 * F;
	tmp = 0.0;
	if (B <= -1.06e+54)
		tmp = sqrt(((-4.0 * (C * F)) + (4.0 * (A * F)))) * (1.0 / B);
	elseif (B <= 18.0)
		tmp = -sqrt((2.0 * (t_1 * (2.0 * A)))) / t_0;
	else
		tmp = -sqrt((2.0 * (t_1 * ((A + C) - B)))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * F), $MachinePrecision]}, If[LessEqual[B, -1.06e+54], N[(N[Sqrt[N[(N[(-4.0 * N[(C * F), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(A * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 18.0], N[((-N[Sqrt[N[(2.0 * N[(t$95$1 * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[((-N[Sqrt[N[(2.0 * N[(t$95$1 * N[(N[(A + C), $MachinePrecision] - B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if B < -1.06e54

   1. Initial program 6.7%

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

   3. Simplified 6.6%

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

   4. Taylor expanded in A around -inf 0.7%

      \[\leadsto \frac{-\sqrt{\color{blue}{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right) + \left(4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right) + -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
   5. Step-by-step derivation

      1. fma-def 0.7%

         \[\leadsto \frac{-\sqrt{\color{blue}{\mathsf{fma}\left(-16, {A}^{2} \cdot \left(C \cdot F\right), 4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right) + -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      2. unpow2 0.7%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \color{blue}{\left(A \cdot A\right)} \cdot \left(C \cdot F\right), 4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right) + -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      3. fma-def 0.7%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \color{blue}{\mathsf{fma}\left(4, A \cdot \left(F \cdot {B}^{2}\right), -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      4. *-commutative 0.7%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \color{blue}{\left({B}^{2} \cdot F\right)}, -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      5. unpow2 0.7%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\color{blue}{\left(B \cdot B\right)} \cdot F\right), -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      6. *-commutative 0.7%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\left(B \cdot B\right) \cdot F\right), -4 \cdot \left(C \cdot \color{blue}{\left({B}^{2} \cdot F\right)}\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      7. unpow2 0.7%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\left(B \cdot B\right) \cdot F\right), -4 \cdot \left(C \cdot \left(\color{blue}{\left(B \cdot B\right)} \cdot F\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   6. Simplified 0.7%

      \[\leadsto \frac{-\sqrt{\color{blue}{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\left(B \cdot B\right) \cdot F\right), -4 \cdot \left(C \cdot \left(\left(B \cdot B\right) \cdot F\right)\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   7. Taylor expanded in B around -inf 6.2%

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

   if -1.06e54 < B < 18

   1. Initial program 26.2%

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

   3. Simplified 26.1%

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

   4. Taylor expanded in A around -inf 29.1%

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

      1. *-commutative 29.1%

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

   6. Simplified 29.1%

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

   if 18 < B

   1. Initial program 20.8%

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

   3. Simplified 20.8%

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

   4. Taylor expanded in B around inf 18.5%

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

4. Final simplification 21.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -1.06 \cdot 10^{+54}:\\ \;\;\;\;\sqrt{-4 \cdot \left(C \cdot F\right) + 4 \cdot \left(A \cdot F\right)} \cdot \frac{1}{B}\\ \mathbf{elif}\;B \leq 18:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(2 \cdot A\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(\left(A + C\right) - B\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \end{array} \]

Alternative 11: 29.9% accurate, 4.8× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := \sqrt{-4 \cdot \left(C \cdot F\right) + 4 \cdot \left(A \cdot F\right)}\\ t_1 := B \cdot B + \left(A \cdot C\right) \cdot -4\\ \mathbf{if}\;B \leq -2.35 \cdot 10^{+55}:\\ \;\;\;\;t_0 \cdot \frac{1}{B}\\ \mathbf{elif}\;B \leq 1.5 \cdot 10^{+28}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{-1}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (sqrt (+ (* -4.0 (* C F)) (* 4.0 (* A F)))))
        (t_1 (+ (* B B) (* (* A C) -4.0))))
   (if (<= B -2.35e+55)
     (* t_0 (/ 1.0 B))
     (if (<= B 1.5e+28)
       (/ (- (sqrt (* 2.0 (* t_1 (* F (* 2.0 A)))))) t_1)
       (* t_0 (/ -1.0 B))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = sqrt(((-4.0 * (C * F)) + (4.0 * (A * F))));
	double t_1 = (B * B) + ((A * C) * -4.0);
	double tmp;
	if (B <= -2.35e+55) {
		tmp = t_0 * (1.0 / B);
	} else if (B <= 1.5e+28) {
		tmp = -sqrt((2.0 * (t_1 * (F * (2.0 * A))))) / t_1;
	} else {
		tmp = t_0 * (-1.0 / B);
	}
	return tmp;
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt((((-4.0d0) * (c * f)) + (4.0d0 * (a * f))))
    t_1 = (b * b) + ((a * c) * (-4.0d0))
    if (b <= (-2.35d+55)) then
        tmp = t_0 * (1.0d0 / b)
    else if (b <= 1.5d+28) then
        tmp = -sqrt((2.0d0 * (t_1 * (f * (2.0d0 * a))))) / t_1
    else
        tmp = t_0 * ((-1.0d0) / b)
    end if
    code = tmp
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = Math.sqrt(((-4.0 * (C * F)) + (4.0 * (A * F))));
	double t_1 = (B * B) + ((A * C) * -4.0);
	double tmp;
	if (B <= -2.35e+55) {
		tmp = t_0 * (1.0 / B);
	} else if (B <= 1.5e+28) {
		tmp = -Math.sqrt((2.0 * (t_1 * (F * (2.0 * A))))) / t_1;
	} else {
		tmp = t_0 * (-1.0 / B);
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = math.sqrt(((-4.0 * (C * F)) + (4.0 * (A * F))))
	t_1 = (B * B) + ((A * C) * -4.0)
	tmp = 0
	if B <= -2.35e+55:
		tmp = t_0 * (1.0 / B)
	elif B <= 1.5e+28:
		tmp = -math.sqrt((2.0 * (t_1 * (F * (2.0 * A))))) / t_1
	else:
		tmp = t_0 * (-1.0 / B)
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = sqrt(Float64(Float64(-4.0 * Float64(C * F)) + Float64(4.0 * Float64(A * F))))
	t_1 = Float64(Float64(B * B) + Float64(Float64(A * C) * -4.0))
	tmp = 0.0
	if (B <= -2.35e+55)
		tmp = Float64(t_0 * Float64(1.0 / B));
	elseif (B <= 1.5e+28)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_1 * Float64(F * Float64(2.0 * A)))))) / t_1);
	else
		tmp = Float64(t_0 * Float64(-1.0 / B));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = sqrt(((-4.0 * (C * F)) + (4.0 * (A * F))));
	t_1 = (B * B) + ((A * C) * -4.0);
	tmp = 0.0;
	if (B <= -2.35e+55)
		tmp = t_0 * (1.0 / B);
	elseif (B <= 1.5e+28)
		tmp = -sqrt((2.0 * (t_1 * (F * (2.0 * A))))) / t_1;
	else
		tmp = t_0 * (-1.0 / B);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[Sqrt[N[(N[(-4.0 * N[(C * F), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(A * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(B * B), $MachinePrecision] + N[(N[(A * C), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -2.35e+55], N[(t$95$0 * N[(1.0 / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 1.5e+28], N[((-N[Sqrt[N[(2.0 * N[(t$95$1 * N[(F * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], N[(t$95$0 * N[(-1.0 / B), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if B < -2.35e55

   1. Initial program 6.7%

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

   3. Simplified 6.6%

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

   4. Taylor expanded in A around -inf 0.7%

      \[\leadsto \frac{-\sqrt{\color{blue}{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right) + \left(4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right) + -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
   5. Step-by-step derivation

      1. fma-def 0.7%

         \[\leadsto \frac{-\sqrt{\color{blue}{\mathsf{fma}\left(-16, {A}^{2} \cdot \left(C \cdot F\right), 4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right) + -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      2. unpow2 0.7%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \color{blue}{\left(A \cdot A\right)} \cdot \left(C \cdot F\right), 4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right) + -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      3. fma-def 0.7%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \color{blue}{\mathsf{fma}\left(4, A \cdot \left(F \cdot {B}^{2}\right), -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      4. *-commutative 0.7%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \color{blue}{\left({B}^{2} \cdot F\right)}, -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      5. unpow2 0.7%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\color{blue}{\left(B \cdot B\right)} \cdot F\right), -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      6. *-commutative 0.7%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\left(B \cdot B\right) \cdot F\right), -4 \cdot \left(C \cdot \color{blue}{\left({B}^{2} \cdot F\right)}\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      7. unpow2 0.7%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\left(B \cdot B\right) \cdot F\right), -4 \cdot \left(C \cdot \left(\color{blue}{\left(B \cdot B\right)} \cdot F\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   6. Simplified 0.7%

      \[\leadsto \frac{-\sqrt{\color{blue}{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\left(B \cdot B\right) \cdot F\right), -4 \cdot \left(C \cdot \left(\left(B \cdot B\right) \cdot F\right)\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   7. Taylor expanded in B around -inf 6.2%

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

   if -2.35e55 < B < 1.5e28

   1. Initial program 28.2%

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

   3. Simplified 28.2%

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

   4. Taylor expanded in A around -inf 29.0%

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

      1. *-commutative 29.0%

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

   6. Simplified 29.0%

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

      1. distribute-frac-neg 29.0%

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

      2. associate-*l* 28.2%

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

      3. cancel-sign-sub-inv 28.2%

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

      4. metadata-eval 28.2%

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

      5. *-commutative 28.2%

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

      6. cancel-sign-sub-inv 28.2%

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

      7. metadata-eval 28.2%

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

   8. Applied egg-rr 28.2%

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

   if 1.5e28 < B

   1. Initial program 14.6%

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

   3. Simplified 11.1%

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

   4. Taylor expanded in A around -inf 0.7%

      \[\leadsto \frac{-\sqrt{\color{blue}{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right) + \left(4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right) + -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
   5. Step-by-step derivation

      1. fma-def 0.7%

         \[\leadsto \frac{-\sqrt{\color{blue}{\mathsf{fma}\left(-16, {A}^{2} \cdot \left(C \cdot F\right), 4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right) + -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      2. unpow2 0.7%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \color{blue}{\left(A \cdot A\right)} \cdot \left(C \cdot F\right), 4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right) + -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      3. fma-def 0.7%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \color{blue}{\mathsf{fma}\left(4, A \cdot \left(F \cdot {B}^{2}\right), -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      4. *-commutative 0.7%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \color{blue}{\left({B}^{2} \cdot F\right)}, -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      5. unpow2 0.7%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\color{blue}{\left(B \cdot B\right)} \cdot F\right), -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      6. *-commutative 0.7%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\left(B \cdot B\right) \cdot F\right), -4 \cdot \left(C \cdot \color{blue}{\left({B}^{2} \cdot F\right)}\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      7. unpow2 0.7%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\left(B \cdot B\right) \cdot F\right), -4 \cdot \left(C \cdot \left(\color{blue}{\left(B \cdot B\right)} \cdot F\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   6. Simplified 0.7%

      \[\leadsto \frac{-\sqrt{\color{blue}{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\left(B \cdot B\right) \cdot F\right), -4 \cdot \left(C \cdot \left(\left(B \cdot B\right) \cdot F\right)\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   7. Taylor expanded in B around inf 4.6%

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{-4 \cdot \left(C \cdot F\right) + 4 \cdot \left(A \cdot F\right)} \cdot \frac{1}{B}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 18.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -2.35 \cdot 10^{+55}:\\ \;\;\;\;\sqrt{-4 \cdot \left(C \cdot F\right) + 4 \cdot \left(A \cdot F\right)} \cdot \frac{1}{B}\\ \mathbf{elif}\;B \leq 1.5 \cdot 10^{+28}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(B \cdot B + \left(A \cdot C\right) \cdot -4\right) \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{B \cdot B + \left(A \cdot C\right) \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-4 \cdot \left(C \cdot F\right) + 4 \cdot \left(A \cdot F\right)} \cdot \frac{-1}{B}\\ \end{array} \]

Alternative 12: 29.7% accurate, 4.8× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ t_1 := \sqrt{-4 \cdot \left(C \cdot F\right) + 4 \cdot \left(A \cdot F\right)}\\ \mathbf{if}\;B \leq -6.4 \cdot 10^{+54}:\\ \;\;\;\;t_1 \cdot \frac{1}{B}\\ \mathbf{elif}\;B \leq 1.3 \cdot 10^{+28}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(t_0 \cdot F\right) \cdot \left(2 \cdot A\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{-1}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* A C))))
        (t_1 (sqrt (+ (* -4.0 (* C F)) (* 4.0 (* A F))))))
   (if (<= B -6.4e+54)
     (* t_1 (/ 1.0 B))
     (if (<= B 1.3e+28)
       (/ (- (sqrt (* 2.0 (* (* t_0 F) (* 2.0 A))))) t_0)
       (* t_1 (/ -1.0 B))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double t_1 = sqrt(((-4.0 * (C * F)) + (4.0 * (A * F))));
	double tmp;
	if (B <= -6.4e+54) {
		tmp = t_1 * (1.0 / B);
	} else if (B <= 1.3e+28) {
		tmp = -sqrt((2.0 * ((t_0 * F) * (2.0 * A)))) / t_0;
	} else {
		tmp = t_1 * (-1.0 / B);
	}
	return tmp;
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (b * b) - (4.0d0 * (a * c))
    t_1 = sqrt((((-4.0d0) * (c * f)) + (4.0d0 * (a * f))))
    if (b <= (-6.4d+54)) then
        tmp = t_1 * (1.0d0 / b)
    else if (b <= 1.3d+28) then
        tmp = -sqrt((2.0d0 * ((t_0 * f) * (2.0d0 * a)))) / t_0
    else
        tmp = t_1 * ((-1.0d0) / b)
    end if
    code = tmp
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double t_1 = Math.sqrt(((-4.0 * (C * F)) + (4.0 * (A * F))));
	double tmp;
	if (B <= -6.4e+54) {
		tmp = t_1 * (1.0 / B);
	} else if (B <= 1.3e+28) {
		tmp = -Math.sqrt((2.0 * ((t_0 * F) * (2.0 * A)))) / t_0;
	} else {
		tmp = t_1 * (-1.0 / B);
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (A * C))
	t_1 = math.sqrt(((-4.0 * (C * F)) + (4.0 * (A * F))))
	tmp = 0
	if B <= -6.4e+54:
		tmp = t_1 * (1.0 / B)
	elif B <= 1.3e+28:
		tmp = -math.sqrt((2.0 * ((t_0 * F) * (2.0 * A)))) / t_0
	else:
		tmp = t_1 * (-1.0 / B)
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	t_1 = sqrt(Float64(Float64(-4.0 * Float64(C * F)) + Float64(4.0 * Float64(A * F))))
	tmp = 0.0
	if (B <= -6.4e+54)
		tmp = Float64(t_1 * Float64(1.0 / B));
	elseif (B <= 1.3e+28)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(t_0 * F) * Float64(2.0 * A))))) / t_0);
	else
		tmp = Float64(t_1 * Float64(-1.0 / B));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (A * C));
	t_1 = sqrt(((-4.0 * (C * F)) + (4.0 * (A * F))));
	tmp = 0.0;
	if (B <= -6.4e+54)
		tmp = t_1 * (1.0 / B);
	elseif (B <= 1.3e+28)
		tmp = -sqrt((2.0 * ((t_0 * F) * (2.0 * A)))) / t_0;
	else
		tmp = t_1 * (-1.0 / B);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(-4.0 * N[(C * F), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(A * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[B, -6.4e+54], N[(t$95$1 * N[(1.0 / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 1.3e+28], N[((-N[Sqrt[N[(2.0 * N[(N[(t$95$0 * F), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(t$95$1 * N[(-1.0 / B), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if B < -6.4e54

   1. Initial program 6.7%

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

   3. Simplified 6.6%

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

   4. Taylor expanded in A around -inf 0.7%

      \[\leadsto \frac{-\sqrt{\color{blue}{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right) + \left(4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right) + -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
   5. Step-by-step derivation

      1. fma-def 0.7%

         \[\leadsto \frac{-\sqrt{\color{blue}{\mathsf{fma}\left(-16, {A}^{2} \cdot \left(C \cdot F\right), 4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right) + -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      2. unpow2 0.7%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \color{blue}{\left(A \cdot A\right)} \cdot \left(C \cdot F\right), 4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right) + -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      3. fma-def 0.7%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \color{blue}{\mathsf{fma}\left(4, A \cdot \left(F \cdot {B}^{2}\right), -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      4. *-commutative 0.7%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \color{blue}{\left({B}^{2} \cdot F\right)}, -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      5. unpow2 0.7%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\color{blue}{\left(B \cdot B\right)} \cdot F\right), -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      6. *-commutative 0.7%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\left(B \cdot B\right) \cdot F\right), -4 \cdot \left(C \cdot \color{blue}{\left({B}^{2} \cdot F\right)}\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      7. unpow2 0.7%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\left(B \cdot B\right) \cdot F\right), -4 \cdot \left(C \cdot \left(\color{blue}{\left(B \cdot B\right)} \cdot F\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   6. Simplified 0.7%

      \[\leadsto \frac{-\sqrt{\color{blue}{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\left(B \cdot B\right) \cdot F\right), -4 \cdot \left(C \cdot \left(\left(B \cdot B\right) \cdot F\right)\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   7. Taylor expanded in B around -inf 6.2%

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

   if -6.4e54 < B < 1.3000000000000001e28

   1. Initial program 28.2%

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

   3. Simplified 28.2%

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

   4. Taylor expanded in A around -inf 29.0%

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

      1. *-commutative 29.0%

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

   6. Simplified 29.0%

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

   if 1.3000000000000001e28 < B

   1. Initial program 14.6%

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

   3. Simplified 11.1%

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

   4. Taylor expanded in A around -inf 0.7%

      \[\leadsto \frac{-\sqrt{\color{blue}{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right) + \left(4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right) + -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
   5. Step-by-step derivation

      1. fma-def 0.7%

         \[\leadsto \frac{-\sqrt{\color{blue}{\mathsf{fma}\left(-16, {A}^{2} \cdot \left(C \cdot F\right), 4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right) + -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      2. unpow2 0.7%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \color{blue}{\left(A \cdot A\right)} \cdot \left(C \cdot F\right), 4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right) + -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      3. fma-def 0.7%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \color{blue}{\mathsf{fma}\left(4, A \cdot \left(F \cdot {B}^{2}\right), -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      4. *-commutative 0.7%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \color{blue}{\left({B}^{2} \cdot F\right)}, -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      5. unpow2 0.7%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\color{blue}{\left(B \cdot B\right)} \cdot F\right), -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      6. *-commutative 0.7%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\left(B \cdot B\right) \cdot F\right), -4 \cdot \left(C \cdot \color{blue}{\left({B}^{2} \cdot F\right)}\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      7. unpow2 0.7%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\left(B \cdot B\right) \cdot F\right), -4 \cdot \left(C \cdot \left(\color{blue}{\left(B \cdot B\right)} \cdot F\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   6. Simplified 0.7%

      \[\leadsto \frac{-\sqrt{\color{blue}{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\left(B \cdot B\right) \cdot F\right), -4 \cdot \left(C \cdot \left(\left(B \cdot B\right) \cdot F\right)\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   7. Taylor expanded in B around inf 4.6%

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{-4 \cdot \left(C \cdot F\right) + 4 \cdot \left(A \cdot F\right)} \cdot \frac{1}{B}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 18.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -6.4 \cdot 10^{+54}:\\ \;\;\;\;\sqrt{-4 \cdot \left(C \cdot F\right) + 4 \cdot \left(A \cdot F\right)} \cdot \frac{1}{B}\\ \mathbf{elif}\;B \leq 1.3 \cdot 10^{+28}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(2 \cdot A\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-4 \cdot \left(C \cdot F\right) + 4 \cdot \left(A \cdot F\right)} \cdot \frac{-1}{B}\\ \end{array} \]

Alternative 13: 21.2% accurate, 5.0× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} \mathbf{if}\;B \leq -1.5 \cdot 10^{-50}:\\ \;\;\;\;\sqrt{-4 \cdot \left(C \cdot F\right) + 4 \cdot \left(A \cdot F\right)} \cdot \frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(2 \cdot A\right) \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (if (<= B -1.5e-50)
   (* (sqrt (+ (* -4.0 (* C F)) (* 4.0 (* A F)))) (/ 1.0 B))
   (/
    (- (sqrt (* 2.0 (* (* 2.0 A) (* -4.0 (* A (* C F)))))))
    (- (* B B) (* 4.0 (* A C))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double tmp;
	if (B <= -1.5e-50) {
		tmp = sqrt(((-4.0 * (C * F)) + (4.0 * (A * F)))) * (1.0 / B);
	} else {
		tmp = -sqrt((2.0 * ((2.0 * A) * (-4.0 * (A * (C * F)))))) / ((B * B) - (4.0 * (A * C)));
	}
	return tmp;
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b <= (-1.5d-50)) then
        tmp = sqrt((((-4.0d0) * (c * f)) + (4.0d0 * (a * f)))) * (1.0d0 / b)
    else
        tmp = -sqrt((2.0d0 * ((2.0d0 * a) * ((-4.0d0) * (a * (c * f)))))) / ((b * b) - (4.0d0 * (a * c)))
    end if
    code = tmp
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double tmp;
	if (B <= -1.5e-50) {
		tmp = Math.sqrt(((-4.0 * (C * F)) + (4.0 * (A * F)))) * (1.0 / B);
	} else {
		tmp = -Math.sqrt((2.0 * ((2.0 * A) * (-4.0 * (A * (C * F)))))) / ((B * B) - (4.0 * (A * C)));
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	tmp = 0
	if B <= -1.5e-50:
		tmp = math.sqrt(((-4.0 * (C * F)) + (4.0 * (A * F)))) * (1.0 / B)
	else:
		tmp = -math.sqrt((2.0 * ((2.0 * A) * (-4.0 * (A * (C * F)))))) / ((B * B) - (4.0 * (A * C)))
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	tmp = 0.0
	if (B <= -1.5e-50)
		tmp = Float64(sqrt(Float64(Float64(-4.0 * Float64(C * F)) + Float64(4.0 * Float64(A * F)))) * Float64(1.0 / B));
	else
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(2.0 * A) * Float64(-4.0 * Float64(A * Float64(C * F))))))) / Float64(Float64(B * B) - Float64(4.0 * Float64(A * C))));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (B <= -1.5e-50)
		tmp = sqrt(((-4.0 * (C * F)) + (4.0 * (A * F)))) * (1.0 / B);
	else
		tmp = -sqrt((2.0 * ((2.0 * A) * (-4.0 * (A * (C * F)))))) / ((B * B) - (4.0 * (A * C)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := If[LessEqual[B, -1.5e-50], N[(N[Sqrt[N[(N[(-4.0 * N[(C * F), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(A * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 / B), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[N[(2.0 * N[(N[(2.0 * A), $MachinePrecision] * N[(-4.0 * N[(A * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < -1.49999999999999995e-50

   1. Initial program 13.5%

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

   3. Simplified 14.1%

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

   4. Taylor expanded in A around -inf 1.0%

      \[\leadsto \frac{-\sqrt{\color{blue}{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right) + \left(4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right) + -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
   5. Step-by-step derivation

      1. fma-def 1.0%

         \[\leadsto \frac{-\sqrt{\color{blue}{\mathsf{fma}\left(-16, {A}^{2} \cdot \left(C \cdot F\right), 4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right) + -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      2. unpow2 1.0%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \color{blue}{\left(A \cdot A\right)} \cdot \left(C \cdot F\right), 4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right) + -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      3. fma-def 1.0%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \color{blue}{\mathsf{fma}\left(4, A \cdot \left(F \cdot {B}^{2}\right), -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      4. *-commutative 1.0%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \color{blue}{\left({B}^{2} \cdot F\right)}, -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      5. unpow2 1.0%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\color{blue}{\left(B \cdot B\right)} \cdot F\right), -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      6. *-commutative 1.0%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\left(B \cdot B\right) \cdot F\right), -4 \cdot \left(C \cdot \color{blue}{\left({B}^{2} \cdot F\right)}\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      7. unpow2 1.0%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\left(B \cdot B\right) \cdot F\right), -4 \cdot \left(C \cdot \left(\color{blue}{\left(B \cdot B\right)} \cdot F\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   6. Simplified 1.0%

      \[\leadsto \frac{-\sqrt{\color{blue}{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\left(B \cdot B\right) \cdot F\right), -4 \cdot \left(C \cdot \left(\left(B \cdot B\right) \cdot F\right)\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   7. Taylor expanded in B around -inf 5.0%

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

   if -1.49999999999999995e-50 < B

   1. Initial program 23.7%

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

   3. Simplified 23.7%

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

   4. Taylor expanded in A around -inf 21.7%

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

      1. *-commutative 21.7%

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

   6. Simplified 21.7%

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

   7. Taylor expanded in B around 0 19.2%

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

4. Final simplification 15.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -1.5 \cdot 10^{-50}:\\ \;\;\;\;\sqrt{-4 \cdot \left(C \cdot F\right) + 4 \cdot \left(A \cdot F\right)} \cdot \frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(2 \cdot A\right) \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \end{array} \]

Alternative 14: 9.6% accurate, 5.4× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} \mathbf{if}\;B \leq 1.75 \cdot 10^{-287}:\\ \;\;\;\;\sqrt{-4 \cdot \left(C \cdot F\right) + 4 \cdot \left(A \cdot F\right)} \cdot \frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{A \cdot F}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (if (<= B 1.75e-287)
   (* (sqrt (+ (* -4.0 (* C F)) (* 4.0 (* A F)))) (/ 1.0 B))
   (* -2.0 (/ (sqrt (* A F)) B))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double tmp;
	if (B <= 1.75e-287) {
		tmp = sqrt(((-4.0 * (C * F)) + (4.0 * (A * F)))) * (1.0 / B);
	} else {
		tmp = -2.0 * (sqrt((A * F)) / B);
	}
	return tmp;
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b <= 1.75d-287) then
        tmp = sqrt((((-4.0d0) * (c * f)) + (4.0d0 * (a * f)))) * (1.0d0 / b)
    else
        tmp = (-2.0d0) * (sqrt((a * f)) / b)
    end if
    code = tmp
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double tmp;
	if (B <= 1.75e-287) {
		tmp = Math.sqrt(((-4.0 * (C * F)) + (4.0 * (A * F)))) * (1.0 / B);
	} else {
		tmp = -2.0 * (Math.sqrt((A * F)) / B);
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	tmp = 0
	if B <= 1.75e-287:
		tmp = math.sqrt(((-4.0 * (C * F)) + (4.0 * (A * F)))) * (1.0 / B)
	else:
		tmp = -2.0 * (math.sqrt((A * F)) / B)
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	tmp = 0.0
	if (B <= 1.75e-287)
		tmp = Float64(sqrt(Float64(Float64(-4.0 * Float64(C * F)) + Float64(4.0 * Float64(A * F)))) * Float64(1.0 / B));
	else
		tmp = Float64(-2.0 * Float64(sqrt(Float64(A * F)) / B));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (B <= 1.75e-287)
		tmp = sqrt(((-4.0 * (C * F)) + (4.0 * (A * F)))) * (1.0 / B);
	else
		tmp = -2.0 * (sqrt((A * F)) / B);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := If[LessEqual[B, 1.75e-287], N[(N[Sqrt[N[(N[(-4.0 * N[(C * F), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(A * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 / B), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1.75e-287

   1. Initial program 20.5%

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

   3. Simplified 21.7%

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

   4. Taylor expanded in A around -inf 12.7%

      \[\leadsto \frac{-\sqrt{\color{blue}{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right) + \left(4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right) + -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
   5. Step-by-step derivation

      1. fma-def 12.7%

         \[\leadsto \frac{-\sqrt{\color{blue}{\mathsf{fma}\left(-16, {A}^{2} \cdot \left(C \cdot F\right), 4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right) + -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      2. unpow2 12.7%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \color{blue}{\left(A \cdot A\right)} \cdot \left(C \cdot F\right), 4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right) + -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      3. fma-def 12.7%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \color{blue}{\mathsf{fma}\left(4, A \cdot \left(F \cdot {B}^{2}\right), -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      4. *-commutative 12.7%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \color{blue}{\left({B}^{2} \cdot F\right)}, -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      5. unpow2 12.7%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\color{blue}{\left(B \cdot B\right)} \cdot F\right), -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      6. *-commutative 12.7%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\left(B \cdot B\right) \cdot F\right), -4 \cdot \left(C \cdot \color{blue}{\left({B}^{2} \cdot F\right)}\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      7. unpow2 12.7%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\left(B \cdot B\right) \cdot F\right), -4 \cdot \left(C \cdot \left(\color{blue}{\left(B \cdot B\right)} \cdot F\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   6. Simplified 12.7%

      \[\leadsto \frac{-\sqrt{\color{blue}{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\left(B \cdot B\right) \cdot F\right), -4 \cdot \left(C \cdot \left(\left(B \cdot B\right) \cdot F\right)\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   7. Taylor expanded in B around -inf 3.6%

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

   if 1.75e-287 < B

   1. Initial program 21.0%

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

   3. Simplified 21.8%

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

   4. Taylor expanded in A around -inf 10.1%

      \[\leadsto \frac{-\sqrt{\color{blue}{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right) + \left(4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right) + -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
   5. Step-by-step derivation

      1. fma-def 10.1%

         \[\leadsto \frac{-\sqrt{\color{blue}{\mathsf{fma}\left(-16, {A}^{2} \cdot \left(C \cdot F\right), 4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right) + -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      2. unpow2 10.1%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \color{blue}{\left(A \cdot A\right)} \cdot \left(C \cdot F\right), 4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right) + -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      3. fma-def 10.1%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \color{blue}{\mathsf{fma}\left(4, A \cdot \left(F \cdot {B}^{2}\right), -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      4. *-commutative 10.1%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \color{blue}{\left({B}^{2} \cdot F\right)}, -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      5. unpow2 10.1%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\color{blue}{\left(B \cdot B\right)} \cdot F\right), -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      6. *-commutative 10.1%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\left(B \cdot B\right) \cdot F\right), -4 \cdot \left(C \cdot \color{blue}{\left({B}^{2} \cdot F\right)}\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      7. unpow2 10.1%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\left(B \cdot B\right) \cdot F\right), -4 \cdot \left(C \cdot \left(\color{blue}{\left(B \cdot B\right)} \cdot F\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   6. Simplified 10.1%

      \[\leadsto \frac{-\sqrt{\color{blue}{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\left(B \cdot B\right) \cdot F\right), -4 \cdot \left(C \cdot \left(\left(B \cdot B\right) \cdot F\right)\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   7. Taylor expanded in C around 0 4.6%

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

      1. associate-*r/ 4.6%

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

      2. *-rgt-identity 4.6%

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

      3. *-commutative 4.6%

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

   9. Simplified 4.6%

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

4. Final simplification 4.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.75 \cdot 10^{-287}:\\ \;\;\;\;\sqrt{-4 \cdot \left(C \cdot F\right) + 4 \cdot \left(A \cdot F\right)} \cdot \frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{A \cdot F}}{B}\\ \end{array} \]

Alternative 15: 9.9% accurate, 5.4× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := \sqrt{-4 \cdot \left(C \cdot F\right) + 4 \cdot \left(A \cdot F\right)}\\ \mathbf{if}\;B \leq 1.45 \cdot 10^{-287}:\\ \;\;\;\;t_0 \cdot \frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{-1}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (sqrt (+ (* -4.0 (* C F)) (* 4.0 (* A F))))))
   (if (<= B 1.45e-287) (* t_0 (/ 1.0 B)) (* t_0 (/ -1.0 B)))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = sqrt(((-4.0 * (C * F)) + (4.0 * (A * F))));
	double tmp;
	if (B <= 1.45e-287) {
		tmp = t_0 * (1.0 / B);
	} else {
		tmp = t_0 * (-1.0 / B);
	}
	return tmp;
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((((-4.0d0) * (c * f)) + (4.0d0 * (a * f))))
    if (b <= 1.45d-287) then
        tmp = t_0 * (1.0d0 / b)
    else
        tmp = t_0 * ((-1.0d0) / b)
    end if
    code = tmp
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = Math.sqrt(((-4.0 * (C * F)) + (4.0 * (A * F))));
	double tmp;
	if (B <= 1.45e-287) {
		tmp = t_0 * (1.0 / B);
	} else {
		tmp = t_0 * (-1.0 / B);
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = math.sqrt(((-4.0 * (C * F)) + (4.0 * (A * F))))
	tmp = 0
	if B <= 1.45e-287:
		tmp = t_0 * (1.0 / B)
	else:
		tmp = t_0 * (-1.0 / B)
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = sqrt(Float64(Float64(-4.0 * Float64(C * F)) + Float64(4.0 * Float64(A * F))))
	tmp = 0.0
	if (B <= 1.45e-287)
		tmp = Float64(t_0 * Float64(1.0 / B));
	else
		tmp = Float64(t_0 * Float64(-1.0 / B));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = sqrt(((-4.0 * (C * F)) + (4.0 * (A * F))));
	tmp = 0.0;
	if (B <= 1.45e-287)
		tmp = t_0 * (1.0 / B);
	else
		tmp = t_0 * (-1.0 / B);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[Sqrt[N[(N[(-4.0 * N[(C * F), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(A * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[B, 1.45e-287], N[(t$95$0 * N[(1.0 / B), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(-1.0 / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 1.4499999999999999e-287

   1. Initial program 20.5%

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

   3. Simplified 21.7%

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

   4. Taylor expanded in A around -inf 12.7%

      \[\leadsto \frac{-\sqrt{\color{blue}{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right) + \left(4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right) + -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
   5. Step-by-step derivation

      1. fma-def 12.7%

         \[\leadsto \frac{-\sqrt{\color{blue}{\mathsf{fma}\left(-16, {A}^{2} \cdot \left(C \cdot F\right), 4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right) + -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      2. unpow2 12.7%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \color{blue}{\left(A \cdot A\right)} \cdot \left(C \cdot F\right), 4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right) + -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      3. fma-def 12.7%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \color{blue}{\mathsf{fma}\left(4, A \cdot \left(F \cdot {B}^{2}\right), -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      4. *-commutative 12.7%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \color{blue}{\left({B}^{2} \cdot F\right)}, -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      5. unpow2 12.7%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\color{blue}{\left(B \cdot B\right)} \cdot F\right), -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      6. *-commutative 12.7%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\left(B \cdot B\right) \cdot F\right), -4 \cdot \left(C \cdot \color{blue}{\left({B}^{2} \cdot F\right)}\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      7. unpow2 12.7%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\left(B \cdot B\right) \cdot F\right), -4 \cdot \left(C \cdot \left(\color{blue}{\left(B \cdot B\right)} \cdot F\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   6. Simplified 12.7%

      \[\leadsto \frac{-\sqrt{\color{blue}{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\left(B \cdot B\right) \cdot F\right), -4 \cdot \left(C \cdot \left(\left(B \cdot B\right) \cdot F\right)\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   7. Taylor expanded in B around -inf 3.6%

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

   if 1.4499999999999999e-287 < B

   1. Initial program 21.0%

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

   3. Simplified 21.8%

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

   4. Taylor expanded in A around -inf 10.1%

      \[\leadsto \frac{-\sqrt{\color{blue}{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right) + \left(4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right) + -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
   5. Step-by-step derivation

      1. fma-def 10.1%

         \[\leadsto \frac{-\sqrt{\color{blue}{\mathsf{fma}\left(-16, {A}^{2} \cdot \left(C \cdot F\right), 4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right) + -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      2. unpow2 10.1%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \color{blue}{\left(A \cdot A\right)} \cdot \left(C \cdot F\right), 4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right) + -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      3. fma-def 10.1%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \color{blue}{\mathsf{fma}\left(4, A \cdot \left(F \cdot {B}^{2}\right), -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      4. *-commutative 10.1%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \color{blue}{\left({B}^{2} \cdot F\right)}, -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      5. unpow2 10.1%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\color{blue}{\left(B \cdot B\right)} \cdot F\right), -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      6. *-commutative 10.1%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\left(B \cdot B\right) \cdot F\right), -4 \cdot \left(C \cdot \color{blue}{\left({B}^{2} \cdot F\right)}\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      7. unpow2 10.1%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\left(B \cdot B\right) \cdot F\right), -4 \cdot \left(C \cdot \left(\color{blue}{\left(B \cdot B\right)} \cdot F\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   6. Simplified 10.1%

      \[\leadsto \frac{-\sqrt{\color{blue}{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\left(B \cdot B\right) \cdot F\right), -4 \cdot \left(C \cdot \left(\left(B \cdot B\right) \cdot F\right)\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   7. Taylor expanded in B around inf 4.5%

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{-4 \cdot \left(C \cdot F\right) + 4 \cdot \left(A \cdot F\right)} \cdot \frac{1}{B}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 4.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.45 \cdot 10^{-287}:\\ \;\;\;\;\sqrt{-4 \cdot \left(C \cdot F\right) + 4 \cdot \left(A \cdot F\right)} \cdot \frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-4 \cdot \left(C \cdot F\right) + 4 \cdot \left(A \cdot F\right)} \cdot \frac{-1}{B}\\ \end{array} \]

Alternative 16: 9.3% accurate, 5.8× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := \sqrt{A \cdot F}\\ \mathbf{if}\;B \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t_0 \cdot \frac{2}{B}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{t_0}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (sqrt (* A F))))
   (if (<= B -5e-310) (* t_0 (/ 2.0 B)) (* -2.0 (/ t_0 B)))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = sqrt((A * F));
	double tmp;
	if (B <= -5e-310) {
		tmp = t_0 * (2.0 / B);
	} else {
		tmp = -2.0 * (t_0 / B);
	}
	return tmp;
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((a * f))
    if (b <= (-5d-310)) then
        tmp = t_0 * (2.0d0 / b)
    else
        tmp = (-2.0d0) * (t_0 / b)
    end if
    code = tmp
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = Math.sqrt((A * F));
	double tmp;
	if (B <= -5e-310) {
		tmp = t_0 * (2.0 / B);
	} else {
		tmp = -2.0 * (t_0 / B);
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = math.sqrt((A * F))
	tmp = 0
	if B <= -5e-310:
		tmp = t_0 * (2.0 / B)
	else:
		tmp = -2.0 * (t_0 / B)
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = sqrt(Float64(A * F))
	tmp = 0.0
	if (B <= -5e-310)
		tmp = Float64(t_0 * Float64(2.0 / B));
	else
		tmp = Float64(-2.0 * Float64(t_0 / B));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = sqrt((A * F));
	tmp = 0.0;
	if (B <= -5e-310)
		tmp = t_0 * (2.0 / B);
	else
		tmp = -2.0 * (t_0 / B);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[B, -5e-310], N[(t$95$0 * N[(2.0 / B), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(t$95$0 / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < -4.999999999999985e-310

   1. Initial program 21.1%

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

   3. Simplified 21.0%

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

   4. Taylor expanded in A around -inf 20.9%

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

      1. *-commutative 20.9%

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

   6. Simplified 20.9%

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

      1. sqrt-prod 20.8%

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

      2. associate-*l* 20.1%

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

      3. cancel-sign-sub-inv 20.1%

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

      4. metadata-eval 20.1%

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

      5. *-commutative 20.1%

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

   8. Applied egg-rr 20.1%

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

   9. Taylor expanded in B around -inf 3.8%

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

       1. *-commutative 3.8%

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

       2. unpow2 3.8%

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

       3. rem-square-sqrt 3.8%

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

   11. Simplified 3.8%

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

   if -4.999999999999985e-310 < B

   1. Initial program 20.4%

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

   3. Simplified 21.2%

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

   4. Taylor expanded in A around -inf 9.8%

      \[\leadsto \frac{-\sqrt{\color{blue}{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right) + \left(4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right) + -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
   5. Step-by-step derivation

      1. fma-def 9.8%

         \[\leadsto \frac{-\sqrt{\color{blue}{\mathsf{fma}\left(-16, {A}^{2} \cdot \left(C \cdot F\right), 4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right) + -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      2. unpow2 9.8%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \color{blue}{\left(A \cdot A\right)} \cdot \left(C \cdot F\right), 4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right) + -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      3. fma-def 9.8%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \color{blue}{\mathsf{fma}\left(4, A \cdot \left(F \cdot {B}^{2}\right), -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      4. *-commutative 9.8%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \color{blue}{\left({B}^{2} \cdot F\right)}, -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      5. unpow2 9.8%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\color{blue}{\left(B \cdot B\right)} \cdot F\right), -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      6. *-commutative 9.8%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\left(B \cdot B\right) \cdot F\right), -4 \cdot \left(C \cdot \color{blue}{\left({B}^{2} \cdot F\right)}\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      7. unpow2 9.8%

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\left(B \cdot B\right) \cdot F\right), -4 \cdot \left(C \cdot \left(\color{blue}{\left(B \cdot B\right)} \cdot F\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   6. Simplified 9.8%

      \[\leadsto \frac{-\sqrt{\color{blue}{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\left(B \cdot B\right) \cdot F\right), -4 \cdot \left(C \cdot \left(\left(B \cdot B\right) \cdot F\right)\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   7. Taylor expanded in C around 0 4.4%

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

      1. associate-*r/ 4.5%

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

      2. *-rgt-identity 4.5%

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

      3. *-commutative 4.5%

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

   9. Simplified 4.5%

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

4. Final simplification 4.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{A \cdot F} \cdot \frac{2}{B}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{A \cdot F}}{B}\\ \end{array} \]

Alternative 17: 5.5% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.8%

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

3. Simplified 21.7%

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

4. Taylor expanded in A around -inf 11.4%

   \[\leadsto \frac{-\sqrt{\color{blue}{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right) + \left(4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right) + -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
5. Step-by-step derivation

   1. fma-def 11.4%

      \[\leadsto \frac{-\sqrt{\color{blue}{\mathsf{fma}\left(-16, {A}^{2} \cdot \left(C \cdot F\right), 4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right) + -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   2. unpow2 11.4%

      \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \color{blue}{\left(A \cdot A\right)} \cdot \left(C \cdot F\right), 4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right) + -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   3. fma-def 11.4%

      \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \color{blue}{\mathsf{fma}\left(4, A \cdot \left(F \cdot {B}^{2}\right), -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   4. *-commutative 11.4%

      \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \color{blue}{\left({B}^{2} \cdot F\right)}, -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   5. unpow2 11.4%

      \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\color{blue}{\left(B \cdot B\right)} \cdot F\right), -4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   6. *-commutative 11.4%

      \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\left(B \cdot B\right) \cdot F\right), -4 \cdot \left(C \cdot \color{blue}{\left({B}^{2} \cdot F\right)}\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   7. unpow2 11.4%

      \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\left(B \cdot B\right) \cdot F\right), -4 \cdot \left(C \cdot \left(\color{blue}{\left(B \cdot B\right)} \cdot F\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

6. Simplified 11.4%

   \[\leadsto \frac{-\sqrt{\color{blue}{\mathsf{fma}\left(-16, \left(A \cdot A\right) \cdot \left(C \cdot F\right), \mathsf{fma}\left(4, A \cdot \left(\left(B \cdot B\right) \cdot F\right), -4 \cdot \left(C \cdot \left(\left(B \cdot B\right) \cdot F\right)\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

7. Taylor expanded in C around 0 2.6%

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

   1. associate-*r/ 2.6%

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

   2. *-rgt-identity 2.6%

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

   3. *-commutative 2.6%

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

9. Simplified 2.6%

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

10. Final simplification 2.6%

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

Reproduce

herbie shell --seed 2023196 
(FPCore (A B C F)
  :name "ABCF->ab-angle b"
  :precision binary64
  (/ (- (sqrt (* (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F)) (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))) (- (pow B 2.0) (* (* 4.0 A) C))))