ABCF->ab-angle b

Percentage Accurate: 19.5% → 49.9%

Time: 26.8s

Alternatives: 16

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: 19.5% 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: 49.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} B = |B|\\ [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := -\frac{\sqrt{\left(2 \cdot \left(F \cdot \left(t_1 - {B}^{2}\right)\right)\right) \cdot \left(\sqrt{{B}^{2} + {\left(A - C\right)}^{2}} - \left(A + C\right)\right)}}{{B}^{2} - t_1}\\ \mathbf{if}\;t_2 \leq -5 \cdot 10^{-202}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(A + \left(C - \mathsf{hypot}\left(A - C, B\right)\right)\right)\right)} \cdot \left(-\sqrt{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \mathsf{fma}\left(-0.5, \frac{B \cdot B}{C}, A\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* A C))))
        (t_1 (* (* 4.0 A) C))
        (t_2
         (-
          (/
           (sqrt
            (*
             (* 2.0 (* F (- t_1 (pow B 2.0))))
             (- (sqrt (+ (pow B 2.0) (pow (- A C) 2.0))) (+ A C))))
           (- (pow B 2.0) t_1)))))
   (if (<= t_2 -5e-202)
     (/
      (*
       (sqrt (* 2.0 (* F (+ A (- C (hypot (- A C) B))))))
       (- (sqrt (fma B B (* C (* A -4.0))))))
      (fma B B (* A (* C -4.0))))
     (if (<= t_2 INFINITY)
       (-
        (/ (sqrt (* 2.0 (* (* F t_0) (+ A (fma -0.5 (/ (* B B) C) A))))) t_0))
       (- (/ (sqrt (* (* 2.0 F) (- A (hypot B A)))) B))))))

C

B = abs(B);
assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double t_1 = (4.0 * A) * C;
	double t_2 = -(sqrt(((2.0 * (F * (t_1 - pow(B, 2.0)))) * (sqrt((pow(B, 2.0) + pow((A - C), 2.0))) - (A + C)))) / (pow(B, 2.0) - t_1));
	double tmp;
	if (t_2 <= -5e-202) {
		tmp = (sqrt((2.0 * (F * (A + (C - hypot((A - C), B)))))) * -sqrt(fma(B, B, (C * (A * -4.0))))) / fma(B, B, (A * (C * -4.0)));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = -(sqrt((2.0 * ((F * t_0) * (A + fma(-0.5, ((B * B) / C), A))))) / t_0);
	} else {
		tmp = -(sqrt(((2.0 * F) * (A - hypot(B, A)))) / B);
	}
	return tmp;
}

Julia

B = abs(B)
A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	t_1 = Float64(Float64(4.0 * A) * C)
	t_2 = Float64(-Float64(sqrt(Float64(Float64(2.0 * Float64(F * Float64(t_1 - (B ^ 2.0)))) * Float64(sqrt(Float64((B ^ 2.0) + (Float64(A - C) ^ 2.0))) - Float64(A + C)))) / Float64((B ^ 2.0) - t_1)))
	tmp = 0.0
	if (t_2 <= -5e-202)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(F * Float64(A + Float64(C - hypot(Float64(A - C), B)))))) * Float64(-sqrt(fma(B, B, Float64(C * Float64(A * -4.0)))))) / fma(B, B, Float64(A * Float64(C * -4.0))));
	elseif (t_2 <= Inf)
		tmp = Float64(-Float64(sqrt(Float64(2.0 * Float64(Float64(F * t_0) * Float64(A + fma(-0.5, Float64(Float64(B * B) / C), A))))) / t_0));
	else
		tmp = Float64(-Float64(sqrt(Float64(Float64(2.0 * F) * Float64(A - hypot(B, A)))) / B));
	end
	return tmp
end

Wolfram

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$2 = (-N[(N[Sqrt[N[(N[(2.0 * N[(F * N[(t$95$1 - N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(N[Power[B, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(A + C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[Power[B, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision])}, If[LessEqual[t$95$2, -5e-202], N[(N[(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] * (-N[Sqrt[N[(B * B + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / N[(B * B + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], (-N[(N[Sqrt[N[(2.0 * N[(N[(F * t$95$0), $MachinePrecision] * N[(A + N[(-0.5 * N[(N[(B * B), $MachinePrecision] / C), $MachinePrecision] + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), (-N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(A - N[Sqrt[B ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision])]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < -4.99999999999999973e-202

   1. Initial program 37.4%

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

   3. Simplified 44.6%

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

      1. sqrt-prod 58.2%

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

      2. associate-*r* 58.2%

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

      3. *-commutative 58.2%

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

      4. associate-*l* 58.2%

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

      5. associate--r- 57.8%

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

      6. +-commutative 57.8%

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

   5. Applied egg-rr 57.8%

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

      1. hypot-def 48.3%

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

      2. unpow2 48.3%

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

      3. unpow2 48.3%

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

      4. +-commutative 48.3%

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

      5. unpow2 48.3%

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

      6. unpow2 48.3%

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

      7. hypot-def 57.8%

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

   7. Simplified 57.8%

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

   if -4.99999999999999973e-202 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < +inf.0

   1. Initial program 12.6%

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

   3. Simplified 12.6%

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

   4. Taylor expanded in C around inf 31.9%

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

      1. associate--l+ 31.9%

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

      2. fma-neg 31.9%

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

      3. associate--l+ 31.9%

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

      4. unpow2 31.9%

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

      5. unpow2 31.9%

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

      6. unpow2 31.9%

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

      7. mul-1-neg 31.9%

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

      8. mul-1-neg 31.9%

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

      9. sqr-neg 31.9%

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

      10. mul-1-neg 31.9%

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

   6. Simplified 31.9%

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

   7. Taylor expanded in A around 0 32.1%

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

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

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in C around 0 1.9%

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

      1. mul-1-neg 1.9%

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

      2. *-commutative 1.9%

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

      3. +-commutative 1.9%

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

      4. unpow2 1.9%

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

      5. unpow2 1.9%

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

      6. hypot-def 21.7%

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

   6. Simplified 21.7%

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

      1. associate-*l/ 21.8%

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

   8. Applied egg-rr 21.8%

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

      1. pow1 21.8%

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

      2. sqrt-unprod 21.8%

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

   10. Applied egg-rr 21.8%

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

       1. unpow1 21.8%

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

       2. associate-*r* 21.8%

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

       3. hypot-def 1.9%

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

       4. unpow2 1.9%

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

       5. unpow2 1.9%

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

       6. +-commutative 1.9%

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

       7. unpow2 1.9%

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

       8. unpow2 1.9%

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

       9. hypot-def 21.8%

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

   12. Simplified 21.8%

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

4. Final simplification 35.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\frac{\sqrt{\left(2 \cdot \left(F \cdot \left(\left(4 \cdot A\right) \cdot C - {B}^{2}\right)\right)\right) \cdot \left(\sqrt{{B}^{2} + {\left(A - C\right)}^{2}} - \left(A + C\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq -5 \cdot 10^{-202}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(A + \left(C - \mathsf{hypot}\left(A - C, B\right)\right)\right)\right)} \cdot \left(-\sqrt{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;-\frac{\sqrt{\left(2 \cdot \left(F \cdot \left(\left(4 \cdot A\right) \cdot C - {B}^{2}\right)\right)\right) \cdot \left(\sqrt{{B}^{2} + {\left(A - C\right)}^{2}} - \left(A + C\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq \infty:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot \left(A + \mathsf{fma}\left(-0.5, \frac{B \cdot B}{C}, A\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}}{B}\\ \end{array} \]

Alternative 2: 45.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} B = |B|\\ [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ t_1 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_2 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq 7.5 \cdot 10^{-99}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \mathsf{fma}\left(-0.5, \frac{B \cdot B}{C}, A\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 1.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot t_1\right)\right) \cdot \left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 7 \cdot 10^{+30}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(F \cdot t_2\right) \cdot \left(A + \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(B, B, 0\right)}{C}, A\right)\right)\right)} \cdot \frac{-1}{t_2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* A C))))
        (t_1 (fma B B (* C (* A -4.0))))
        (t_2 (+ (* B B) (* -4.0 (* A C)))))
   (if (<= B 7.5e-99)
     (- (/ (sqrt (* 2.0 (* (* F t_0) (+ A (fma -0.5 (/ (* B B) C) A))))) t_0))
     (if (<= B 1.5e-5)
       (/ (- (sqrt (* (* 2.0 (* F t_1)) (- A (- (hypot B (- A C)) C))))) t_1)
       (if (<= B 7e+30)
         (*
          (sqrt (* 2.0 (* (* F t_2) (+ A (fma -0.5 (/ (fma B B 0.0) C) A)))))
          (/ -1.0 t_2))
         (- (/ (sqrt (* (* 2.0 F) (- A (hypot B A)))) B)))))))

C

B = abs(B);
assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double t_1 = fma(B, B, (C * (A * -4.0)));
	double t_2 = (B * B) + (-4.0 * (A * C));
	double tmp;
	if (B <= 7.5e-99) {
		tmp = -(sqrt((2.0 * ((F * t_0) * (A + fma(-0.5, ((B * B) / C), A))))) / t_0);
	} else if (B <= 1.5e-5) {
		tmp = -sqrt(((2.0 * (F * t_1)) * (A - (hypot(B, (A - C)) - C)))) / t_1;
	} else if (B <= 7e+30) {
		tmp = sqrt((2.0 * ((F * t_2) * (A + fma(-0.5, (fma(B, B, 0.0) / C), A))))) * (-1.0 / t_2);
	} else {
		tmp = -(sqrt(((2.0 * F) * (A - hypot(B, A)))) / B);
	}
	return tmp;
}

Julia

B = abs(B)
A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	t_1 = fma(B, B, Float64(C * Float64(A * -4.0)))
	t_2 = Float64(Float64(B * B) + Float64(-4.0 * Float64(A * C)))
	tmp = 0.0
	if (B <= 7.5e-99)
		tmp = Float64(-Float64(sqrt(Float64(2.0 * Float64(Float64(F * t_0) * Float64(A + fma(-0.5, Float64(Float64(B * B) / C), A))))) / t_0));
	elseif (B <= 1.5e-5)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(F * t_1)) * Float64(A - Float64(hypot(B, Float64(A - C)) - C))))) / t_1);
	elseif (B <= 7e+30)
		tmp = Float64(sqrt(Float64(2.0 * Float64(Float64(F * t_2) * Float64(A + fma(-0.5, Float64(fma(B, B, 0.0) / C), A))))) * Float64(-1.0 / t_2));
	else
		tmp = Float64(-Float64(sqrt(Float64(Float64(2.0 * F) * Float64(A - hypot(B, A)))) / B));
	end
	return tmp
end

Wolfram

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(B * B + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(B * B), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, 7.5e-99], (-N[(N[Sqrt[N[(2.0 * N[(N[(F * t$95$0), $MachinePrecision] * N[(A + N[(-0.5 * N[(N[(B * B), $MachinePrecision] / C), $MachinePrecision] + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), If[LessEqual[B, 1.5e-5], N[((-N[Sqrt[N[(N[(2.0 * N[(F * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(A - N[(N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision] - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], If[LessEqual[B, 7e+30], N[(N[Sqrt[N[(2.0 * N[(N[(F * t$95$2), $MachinePrecision] * N[(A + N[(-0.5 * N[(N[(B * B + 0.0), $MachinePrecision] / C), $MachinePrecision] + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / t$95$2), $MachinePrecision]), $MachinePrecision], (-N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(A - N[Sqrt[B ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision])]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 7.4999999999999999e-99

   1. Initial program 14.4%

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

   3. Simplified 14.4%

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

   4. Taylor expanded in C around inf 15.5%

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

      1. associate--l+ 15.5%

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

      2. fma-neg 15.5%

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

      3. associate--l+ 14.9%

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

      4. unpow2 14.9%

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

      5. unpow2 14.9%

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

      6. unpow2 14.9%

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

      7. mul-1-neg 14.9%

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

      8. mul-1-neg 14.9%

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

      9. sqr-neg 14.9%

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

      10. mul-1-neg 14.9%

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

   6. Simplified 14.9%

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

   7. Taylor expanded in A around 0 15.8%

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

   if 7.4999999999999999e-99 < B < 1.50000000000000004e-5

   1. Initial program 29.4%

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

   3. Simplified 34.6%

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

   if 1.50000000000000004e-5 < B < 7.00000000000000042e30

   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 C around inf 24.6%

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

      1. associate--l+ 24.6%

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

      2. fma-neg 24.6%

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

      3. associate--l+ 24.6%

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

      4. unpow2 24.6%

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

      5. unpow2 24.6%

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

      6. unpow2 24.6%

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

      7. mul-1-neg 24.6%

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

      8. mul-1-neg 24.6%

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

      9. sqr-neg 24.6%

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

      10. mul-1-neg 24.6%

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

   6. Simplified 24.6%

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

      1. div-inv 25.0%

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

   8. Applied egg-rr 25.0%

      \[\leadsto \color{blue}{\left(-\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right) \cdot \left(A + \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(B, B, 0\right)}{C}, A\right)\right)\right)}\right) \cdot \frac{1}{B \cdot B + -4 \cdot \left(A \cdot C\right)}} \]

   if 7.00000000000000042e30 < B

   1. Initial program 8.5%

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

   3. Simplified 8.5%

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

   4. Taylor expanded in C around 0 16.0%

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

      1. mul-1-neg 16.0%

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

      2. *-commutative 16.0%

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

      3. +-commutative 16.0%

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

      4. unpow2 16.0%

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

      5. unpow2 16.0%

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

      6. hypot-def 51.5%

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

   6. Simplified 51.5%

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

      1. associate-*l/ 51.6%

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

   8. Applied egg-rr 51.6%

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

      1. pow1 51.6%

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

      2. sqrt-unprod 51.7%

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

   10. Applied egg-rr 51.7%

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

       1. unpow1 51.7%

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

       2. associate-*r* 51.7%

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

       3. hypot-def 16.1%

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

       4. unpow2 16.1%

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

       5. unpow2 16.1%

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

       6. +-commutative 16.1%

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

       7. unpow2 16.1%

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

       8. unpow2 16.1%

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

       9. hypot-def 51.7%

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

   12. Simplified 51.7%

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

4. Final simplification 26.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 7.5 \cdot 10^{-99}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot \left(A + \mathsf{fma}\left(-0.5, \frac{B \cdot B}{C}, A\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 1.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)\right) \cdot \left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 7 \cdot 10^{+30}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right) \cdot \left(A + \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(B, B, 0\right)}{C}, A\right)\right)\right)} \cdot \frac{-1}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}}{B}\\ \end{array} \]

Alternative 3: 45.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} B = |B|\\ [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ t_1 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_2 := F \cdot t_0\\ \mathbf{if}\;B \leq 1.08 \cdot 10^{-104}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(t_2 \cdot \left(A + \mathsf{fma}\left(-0.5, \frac{B \cdot B}{C}, A\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 2.5 \cdot 10^{-5}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(t_2 \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 2.7 \cdot 10^{+31}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(F \cdot t_1\right) \cdot \left(A + \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(B, B, 0\right)}{C}, A\right)\right)\right)} \cdot \frac{-1}{t_1}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* A C))))
        (t_1 (+ (* B B) (* -4.0 (* A C))))
        (t_2 (* F t_0)))
   (if (<= B 1.08e-104)
     (- (/ (sqrt (* 2.0 (* t_2 (+ A (fma -0.5 (/ (* B B) C) A))))) t_0))
     (if (<= B 2.5e-5)
       (- (/ (sqrt (* 2.0 (* t_2 (- A (hypot A B))))) t_0))
       (if (<= B 2.7e+31)
         (*
          (sqrt (* 2.0 (* (* F t_1) (+ A (fma -0.5 (/ (fma B B 0.0) C) A)))))
          (/ -1.0 t_1))
         (- (/ (sqrt (* (* 2.0 F) (- A (hypot B A)))) B)))))))

C

B = abs(B);
assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double t_1 = (B * B) + (-4.0 * (A * C));
	double t_2 = F * t_0;
	double tmp;
	if (B <= 1.08e-104) {
		tmp = -(sqrt((2.0 * (t_2 * (A + fma(-0.5, ((B * B) / C), A))))) / t_0);
	} else if (B <= 2.5e-5) {
		tmp = -(sqrt((2.0 * (t_2 * (A - hypot(A, B))))) / t_0);
	} else if (B <= 2.7e+31) {
		tmp = sqrt((2.0 * ((F * t_1) * (A + fma(-0.5, (fma(B, B, 0.0) / C), A))))) * (-1.0 / t_1);
	} else {
		tmp = -(sqrt(((2.0 * F) * (A - hypot(B, A)))) / B);
	}
	return tmp;
}

Julia

B = abs(B)
A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	t_1 = Float64(Float64(B * B) + Float64(-4.0 * Float64(A * C)))
	t_2 = Float64(F * t_0)
	tmp = 0.0
	if (B <= 1.08e-104)
		tmp = Float64(-Float64(sqrt(Float64(2.0 * Float64(t_2 * Float64(A + fma(-0.5, Float64(Float64(B * B) / C), A))))) / t_0));
	elseif (B <= 2.5e-5)
		tmp = Float64(-Float64(sqrt(Float64(2.0 * Float64(t_2 * Float64(A - hypot(A, B))))) / t_0));
	elseif (B <= 2.7e+31)
		tmp = Float64(sqrt(Float64(2.0 * Float64(Float64(F * t_1) * Float64(A + fma(-0.5, Float64(fma(B, B, 0.0) / C), A))))) * Float64(-1.0 / t_1));
	else
		tmp = Float64(-Float64(sqrt(Float64(Float64(2.0 * F) * Float64(A - hypot(B, A)))) / B));
	end
	return tmp
end

Wolfram

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(B * B), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(F * t$95$0), $MachinePrecision]}, If[LessEqual[B, 1.08e-104], (-N[(N[Sqrt[N[(2.0 * N[(t$95$2 * N[(A + N[(-0.5 * N[(N[(B * B), $MachinePrecision] / C), $MachinePrecision] + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), If[LessEqual[B, 2.5e-5], (-N[(N[Sqrt[N[(2.0 * N[(t$95$2 * N[(A - N[Sqrt[A ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), If[LessEqual[B, 2.7e+31], N[(N[Sqrt[N[(2.0 * N[(N[(F * t$95$1), $MachinePrecision] * N[(A + N[(-0.5 * N[(N[(B * B + 0.0), $MachinePrecision] / C), $MachinePrecision] + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / t$95$1), $MachinePrecision]), $MachinePrecision], (-N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(A - N[Sqrt[B ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision])]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 1.07999999999999997e-104

   1. Initial program 14.4%

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

   3. Simplified 14.4%

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

   4. Taylor expanded in C around inf 15.5%

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

      1. associate--l+ 15.5%

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

      2. fma-neg 15.5%

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

      3. associate--l+ 14.9%

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

      4. unpow2 14.9%

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

      5. unpow2 14.9%

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

      6. unpow2 14.9%

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

      7. mul-1-neg 14.9%

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

      8. mul-1-neg 14.9%

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

      9. sqr-neg 14.9%

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

      10. mul-1-neg 14.9%

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

   6. Simplified 14.9%

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

   7. Taylor expanded in A around 0 15.8%

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

   if 1.07999999999999997e-104 < B < 2.50000000000000012e-5

   1. Initial program 29.4%

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

   3. Simplified 29.4%

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

   4. Taylor expanded in C around 0 21.6%

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

      1. +-commutative 21.6%

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

      2. unpow2 21.6%

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

      3. unpow2 21.6%

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

      4. hypot-def 21.7%

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

   6. Simplified 21.7%

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

   if 2.50000000000000012e-5 < B < 2.69999999999999986e31

   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 C around inf 24.6%

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

      1. associate--l+ 24.6%

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

      2. fma-neg 24.6%

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

      3. associate--l+ 24.6%

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

      4. unpow2 24.6%

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

      5. unpow2 24.6%

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

      6. unpow2 24.6%

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

      7. mul-1-neg 24.6%

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

      8. mul-1-neg 24.6%

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

      9. sqr-neg 24.6%

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

      10. mul-1-neg 24.6%

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

   6. Simplified 24.6%

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

      1. div-inv 25.0%

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

   8. Applied egg-rr 25.0%

      \[\leadsto \color{blue}{\left(-\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right) \cdot \left(A + \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(B, B, 0\right)}{C}, A\right)\right)\right)}\right) \cdot \frac{1}{B \cdot B + -4 \cdot \left(A \cdot C\right)}} \]

   if 2.69999999999999986e31 < B

   1. Initial program 8.5%

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

   3. Simplified 8.5%

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

   4. Taylor expanded in C around 0 16.0%

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

      1. mul-1-neg 16.0%

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

      2. *-commutative 16.0%

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

      3. +-commutative 16.0%

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

      4. unpow2 16.0%

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

      5. unpow2 16.0%

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

      6. hypot-def 51.5%

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

   6. Simplified 51.5%

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

      1. associate-*l/ 51.6%

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

   8. Applied egg-rr 51.6%

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

      1. pow1 51.6%

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

      2. sqrt-unprod 51.7%

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

   10. Applied egg-rr 51.7%

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

       1. unpow1 51.7%

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

       2. associate-*r* 51.7%

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

       3. hypot-def 16.1%

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

       4. unpow2 16.1%

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

       5. unpow2 16.1%

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

       6. +-commutative 16.1%

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

       7. unpow2 16.1%

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

       8. unpow2 16.1%

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

       9. hypot-def 51.7%

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

   12. Simplified 51.7%

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

4. Final simplification 25.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.08 \cdot 10^{-104}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot \left(A + \mathsf{fma}\left(-0.5, \frac{B \cdot B}{C}, A\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 2.5 \cdot 10^{-5}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 2.7 \cdot 10^{+31}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right) \cdot \left(A + \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(B, B, 0\right)}{C}, A\right)\right)\right)} \cdot \frac{-1}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}}{B}\\ \end{array} \]

Alternative 4: 45.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} B = |B|\\ [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ t_1 := F \cdot t_0\\ t_2 := -\frac{\sqrt{2 \cdot \left(t_1 \cdot \left(A + \mathsf{fma}\left(-0.5, \frac{B \cdot B}{C}, A\right)\right)\right)}}{t_0}\\ \mathbf{if}\;B \leq 1.6 \cdot 10^{-100}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 0.015:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(t_1 \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 5.6 \cdot 10^{+31}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* A C))))
        (t_1 (* F t_0))
        (t_2
         (- (/ (sqrt (* 2.0 (* t_1 (+ A (fma -0.5 (/ (* B B) C) A))))) t_0))))
   (if (<= B 1.6e-100)
     t_2
     (if (<= B 0.015)
       (- (/ (sqrt (* 2.0 (* t_1 (- A (hypot A B))))) t_0))
       (if (<= B 5.6e+31)
         t_2
         (- (/ (sqrt (* (* 2.0 F) (- A (hypot B A)))) B)))))))

C

B = abs(B);
assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double t_1 = F * t_0;
	double t_2 = -(sqrt((2.0 * (t_1 * (A + fma(-0.5, ((B * B) / C), A))))) / t_0);
	double tmp;
	if (B <= 1.6e-100) {
		tmp = t_2;
	} else if (B <= 0.015) {
		tmp = -(sqrt((2.0 * (t_1 * (A - hypot(A, B))))) / t_0);
	} else if (B <= 5.6e+31) {
		tmp = t_2;
	} else {
		tmp = -(sqrt(((2.0 * F) * (A - hypot(B, A)))) / B);
	}
	return tmp;
}

Julia

B = abs(B)
A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	t_1 = Float64(F * t_0)
	t_2 = Float64(-Float64(sqrt(Float64(2.0 * Float64(t_1 * Float64(A + fma(-0.5, Float64(Float64(B * B) / C), A))))) / t_0))
	tmp = 0.0
	if (B <= 1.6e-100)
		tmp = t_2;
	elseif (B <= 0.015)
		tmp = Float64(-Float64(sqrt(Float64(2.0 * Float64(t_1 * Float64(A - hypot(A, B))))) / t_0));
	elseif (B <= 5.6e+31)
		tmp = t_2;
	else
		tmp = Float64(-Float64(sqrt(Float64(Float64(2.0 * F) * Float64(A - hypot(B, A)))) / B));
	end
	return tmp
end

Wolfram

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(F * t$95$0), $MachinePrecision]}, Block[{t$95$2 = (-N[(N[Sqrt[N[(2.0 * N[(t$95$1 * N[(A + N[(-0.5 * N[(N[(B * B), $MachinePrecision] / C), $MachinePrecision] + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision])}, If[LessEqual[B, 1.6e-100], t$95$2, If[LessEqual[B, 0.015], (-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, 5.6e+31], t$95$2, (-N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(A - N[Sqrt[B ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision])]]]]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.60000000000000008e-100 or 0.014999999999999999 < B < 5.60000000000000034e31

   1. Initial program 14.7%

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

   3. Simplified 14.7%

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

   4. Taylor expanded in C around inf 15.7%

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

      1. associate--l+ 15.7%

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

      2. fma-neg 15.7%

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

      3. associate--l+ 15.1%

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

      4. unpow2 15.1%

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

      5. unpow2 15.1%

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

      6. unpow2 15.1%

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

      7. mul-1-neg 15.1%

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

      8. mul-1-neg 15.1%

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

      9. sqr-neg 15.1%

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

      10. mul-1-neg 15.1%

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

   6. Simplified 15.1%

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

   7. Taylor expanded in A around 0 16.0%

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

   if 1.60000000000000008e-100 < B < 0.014999999999999999

   1. Initial program 29.4%

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

   3. Simplified 29.4%

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

   4. Taylor expanded in C around 0 21.6%

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

      1. +-commutative 21.6%

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

      2. unpow2 21.6%

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

      3. unpow2 21.6%

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

      4. hypot-def 21.7%

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

   6. Simplified 21.7%

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

   if 5.60000000000000034e31 < B

   1. Initial program 8.5%

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

   3. Simplified 8.5%

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

   4. Taylor expanded in C around 0 16.0%

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

      1. mul-1-neg 16.0%

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

      2. *-commutative 16.0%

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

      3. +-commutative 16.0%

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

      4. unpow2 16.0%

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

      5. unpow2 16.0%

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

      6. hypot-def 51.5%

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

   6. Simplified 51.5%

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

      1. associate-*l/ 51.6%

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

   8. Applied egg-rr 51.6%

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

      1. pow1 51.6%

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

      2. sqrt-unprod 51.7%

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

   10. Applied egg-rr 51.7%

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

       1. unpow1 51.7%

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

       2. associate-*r* 51.7%

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

       3. hypot-def 16.1%

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

       4. unpow2 16.1%

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

       5. unpow2 16.1%

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

       6. +-commutative 16.1%

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

       7. unpow2 16.1%

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

       8. unpow2 16.1%

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

       9. hypot-def 51.7%

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

   12. Simplified 51.7%

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

4. Final simplification 25.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.6 \cdot 10^{-100}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot \left(A + \mathsf{fma}\left(-0.5, \frac{B \cdot B}{C}, A\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 0.015:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 5.6 \cdot 10^{+31}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot \left(A + \mathsf{fma}\left(-0.5, \frac{B \cdot B}{C}, A\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}}{B}\\ \end{array} \]

Alternative 5: 45.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} B = |B|\\ [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq 9.5 \cdot 10^{-67}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(2 \cdot A\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot F\right) \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* A C)))))
   (if (<= B 9.5e-67)
     (/ (- (sqrt (* 2.0 (* (* F t_0) (* 2.0 A))))) t_0)
     (/ (- (sqrt (* (* 2.0 F) (- A (hypot B A))))) B))))

C

B = abs(B);
assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (B <= 9.5e-67) {
		tmp = -sqrt((2.0 * ((F * t_0) * (2.0 * A)))) / t_0;
	} else {
		tmp = -sqrt(((2.0 * F) * (A - hypot(B, A)))) / B;
	}
	return tmp;
}

Java

B = Math.abs(B);
assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (B <= 9.5e-67) {
		tmp = -Math.sqrt((2.0 * ((F * t_0) * (2.0 * A)))) / t_0;
	} else {
		tmp = -Math.sqrt(((2.0 * F) * (A - Math.hypot(B, A)))) / B;
	}
	return tmp;
}

Python

B = abs(B)
[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (A * C))
	tmp = 0
	if B <= 9.5e-67:
		tmp = -math.sqrt((2.0 * ((F * t_0) * (2.0 * A)))) / t_0
	else:
		tmp = -math.sqrt(((2.0 * F) * (A - math.hypot(B, A)))) / B
	return tmp

Julia

B = abs(B)
A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	tmp = 0.0
	if (B <= 9.5e-67)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(F * t_0) * Float64(2.0 * A))))) / t_0);
	else
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * F) * Float64(A - hypot(B, A))))) / B);
	end
	return tmp
end

MATLAB

B = abs(B)
A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (A * C));
	tmp = 0.0;
	if (B <= 9.5e-67)
		tmp = -sqrt((2.0 * ((F * t_0) * (2.0 * A)))) / t_0;
	else
		tmp = -sqrt(((2.0 * F) * (A - hypot(B, A)))) / B;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, 9.5e-67], N[((-N[Sqrt[N[(2.0 * N[(N[(F * t$95$0), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[((-N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(A - N[Sqrt[B ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / B), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 9.4999999999999994e-67

   1. Initial program 15.2%

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

   3. Simplified 15.2%

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

   4. Taylor expanded in A around -inf 14.8%

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

      1. *-commutative 14.8%

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

   6. Simplified 14.8%

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

   if 9.4999999999999994e-67 < B

   1. Initial program 12.7%

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

   3. Simplified 12.7%

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

   4. Taylor expanded in C around 0 17.6%

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

      1. mul-1-neg 17.6%

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

      2. *-commutative 17.6%

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

      3. +-commutative 17.6%

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

      4. unpow2 17.6%

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

      5. unpow2 17.6%

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

      6. hypot-def 44.6%

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

   6. Simplified 44.6%

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

      1. associate-*l/ 44.7%

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

   8. Applied egg-rr 44.7%

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

      1. pow1 44.7%

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

      2. sqrt-unprod 44.8%

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

   10. Applied egg-rr 44.8%

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

       1. unpow1 44.8%

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

       2. associate-*r* 44.8%

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

       3. hypot-def 17.7%

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

       4. unpow2 17.7%

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

       5. unpow2 17.7%

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

       6. +-commutative 17.7%

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

       7. unpow2 17.7%

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

       8. unpow2 17.7%

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

       9. hypot-def 44.8%

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

   12. Simplified 44.8%

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

4. Final simplification 25.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 9.5 \cdot 10^{-67}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot \left(2 \cdot A\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot F\right) \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}}{B}\\ \end{array} \]

Alternative 6: 43.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} B = |B|\\ [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq 3.1 \cdot 10^{+37}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(2 \cdot A\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - B\right)}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* A C)))))
   (if (<= B 3.1e+37)
     (/ (- (sqrt (* 2.0 (* (* F t_0) (* 2.0 A))))) t_0)
     (* (/ (sqrt 2.0) B) (- (sqrt (* F (- A B))))))))

C

B = abs(B);
assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (B <= 3.1e+37) {
		tmp = -sqrt((2.0 * ((F * t_0) * (2.0 * A)))) / t_0;
	} else {
		tmp = (sqrt(2.0) / B) * -sqrt((F * (A - B)));
	}
	return tmp;
}

Fortran

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b * b) - (4.0d0 * (a * c))
    if (b <= 3.1d+37) then
        tmp = -sqrt((2.0d0 * ((f * t_0) * (2.0d0 * a)))) / t_0
    else
        tmp = (sqrt(2.0d0) / b) * -sqrt((f * (a - b)))
    end if
    code = tmp
end function

Java

B = Math.abs(B);
assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (B <= 3.1e+37) {
		tmp = -Math.sqrt((2.0 * ((F * t_0) * (2.0 * A)))) / t_0;
	} else {
		tmp = (Math.sqrt(2.0) / B) * -Math.sqrt((F * (A - B)));
	}
	return tmp;
}

Python

B = abs(B)
[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (A * C))
	tmp = 0
	if B <= 3.1e+37:
		tmp = -math.sqrt((2.0 * ((F * t_0) * (2.0 * A)))) / t_0
	else:
		tmp = (math.sqrt(2.0) / B) * -math.sqrt((F * (A - B)))
	return tmp

Julia

B = abs(B)
A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	tmp = 0.0
	if (B <= 3.1e+37)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(F * t_0) * Float64(2.0 * A))))) / t_0);
	else
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(-sqrt(Float64(F * Float64(A - B)))));
	end
	return tmp
end

MATLAB

B = abs(B)
A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (A * C));
	tmp = 0.0;
	if (B <= 3.1e+37)
		tmp = -sqrt((2.0 * ((F * t_0) * (2.0 * A)))) / t_0;
	else
		tmp = (sqrt(2.0) / B) * -sqrt((F * (A - B)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, 3.1e+37], N[((-N[Sqrt[N[(2.0 * N[(N[(F * t$95$0), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * (-N[Sqrt[N[(F * N[(A - B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 3.1000000000000002e37

   1. Initial program 16.3%

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

   3. Simplified 16.3%

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

   4. Taylor expanded in A around -inf 14.0%

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

      1. *-commutative 14.0%

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

   6. Simplified 14.0%

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

   if 3.1000000000000002e37 < B

   1. Initial program 8.6%

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

   3. Simplified 8.6%

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

   4. Taylor expanded in C around 0 16.2%

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

      1. mul-1-neg 16.2%

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

      2. *-commutative 16.2%

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

      3. +-commutative 16.2%

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

      4. unpow2 16.2%

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

      5. unpow2 16.2%

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

      6. hypot-def 52.2%

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

   6. Simplified 52.2%

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

   7. Taylor expanded in A around 0 46.8%

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

      1. mul-1-neg 46.8%

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

      2. unsub-neg 46.8%

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

   9. Simplified 46.8%

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

4. Final simplification 22.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.1 \cdot 10^{+37}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot \left(2 \cdot A\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - B\right)}\right)\\ \end{array} \]

Alternative 7: 42.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} B = |B|\\ [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq 1.75 \cdot 10^{+36}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(2 \cdot A\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{B \cdot \left(-F\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* A C)))))
   (if (<= B 1.75e+36)
     (/ (- (sqrt (* 2.0 (* (* F t_0) (* 2.0 A))))) t_0)
     (* (sqrt (* B (- F))) (/ (- (sqrt 2.0)) B)))))

C

B = abs(B);
assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (B <= 1.75e+36) {
		tmp = -sqrt((2.0 * ((F * t_0) * (2.0 * A)))) / t_0;
	} else {
		tmp = sqrt((B * -F)) * (-sqrt(2.0) / B);
	}
	return tmp;
}

Fortran

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b * b) - (4.0d0 * (a * c))
    if (b <= 1.75d+36) then
        tmp = -sqrt((2.0d0 * ((f * t_0) * (2.0d0 * a)))) / t_0
    else
        tmp = sqrt((b * -f)) * (-sqrt(2.0d0) / b)
    end if
    code = tmp
end function

Java

B = Math.abs(B);
assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (B <= 1.75e+36) {
		tmp = -Math.sqrt((2.0 * ((F * t_0) * (2.0 * A)))) / t_0;
	} else {
		tmp = Math.sqrt((B * -F)) * (-Math.sqrt(2.0) / B);
	}
	return tmp;
}

Python

B = abs(B)
[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (A * C))
	tmp = 0
	if B <= 1.75e+36:
		tmp = -math.sqrt((2.0 * ((F * t_0) * (2.0 * A)))) / t_0
	else:
		tmp = math.sqrt((B * -F)) * (-math.sqrt(2.0) / B)
	return tmp

Julia

B = abs(B)
A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	tmp = 0.0
	if (B <= 1.75e+36)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(F * t_0) * Float64(2.0 * A))))) / t_0);
	else
		tmp = Float64(sqrt(Float64(B * Float64(-F))) * Float64(Float64(-sqrt(2.0)) / B));
	end
	return tmp
end

MATLAB

B = abs(B)
A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (A * C));
	tmp = 0.0;
	if (B <= 1.75e+36)
		tmp = -sqrt((2.0 * ((F * t_0) * (2.0 * A)))) / t_0;
	else
		tmp = sqrt((B * -F)) * (-sqrt(2.0) / B);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, 1.75e+36], N[((-N[Sqrt[N[(2.0 * N[(N[(F * t$95$0), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(N[Sqrt[N[(B * (-F)), $MachinePrecision]], $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 1.7499999999999999e36

   1. Initial program 16.3%

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

   3. Simplified 16.3%

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

   4. Taylor expanded in A around -inf 14.0%

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

      1. *-commutative 14.0%

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

   6. Simplified 14.0%

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

   if 1.7499999999999999e36 < B

   1. Initial program 8.6%

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

   3. Simplified 8.6%

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

   4. Taylor expanded in C around 0 16.2%

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

      1. mul-1-neg 16.2%

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

      2. *-commutative 16.2%

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

      3. +-commutative 16.2%

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

      4. unpow2 16.2%

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

      5. unpow2 16.2%

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

      6. hypot-def 52.2%

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

   6. Simplified 52.2%

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

   7. Taylor expanded in A around 0 46.3%

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

      1. mul-1-neg 46.3%

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

   9. Simplified 46.3%

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

4. Final simplification 22.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.75 \cdot 10^{+36}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot \left(2 \cdot A\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{B \cdot \left(-F\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 8: 42.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} B = |B|\\ [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq 1.55 \cdot 10^{+36}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(2 \cdot A\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{B \cdot \left(-F\right)} \cdot \left(-\sqrt{2}\right)}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* A C)))))
   (if (<= B 1.55e+36)
     (/ (- (sqrt (* 2.0 (* (* F t_0) (* 2.0 A))))) t_0)
     (/ (* (sqrt (* B (- F))) (- (sqrt 2.0))) B))))

C

B = abs(B);
assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (B <= 1.55e+36) {
		tmp = -sqrt((2.0 * ((F * t_0) * (2.0 * A)))) / t_0;
	} else {
		tmp = (sqrt((B * -F)) * -sqrt(2.0)) / B;
	}
	return tmp;
}

Fortran

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b * b) - (4.0d0 * (a * c))
    if (b <= 1.55d+36) then
        tmp = -sqrt((2.0d0 * ((f * t_0) * (2.0d0 * a)))) / t_0
    else
        tmp = (sqrt((b * -f)) * -sqrt(2.0d0)) / b
    end if
    code = tmp
end function

Java

B = Math.abs(B);
assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (B <= 1.55e+36) {
		tmp = -Math.sqrt((2.0 * ((F * t_0) * (2.0 * A)))) / t_0;
	} else {
		tmp = (Math.sqrt((B * -F)) * -Math.sqrt(2.0)) / B;
	}
	return tmp;
}

Python

B = abs(B)
[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (A * C))
	tmp = 0
	if B <= 1.55e+36:
		tmp = -math.sqrt((2.0 * ((F * t_0) * (2.0 * A)))) / t_0
	else:
		tmp = (math.sqrt((B * -F)) * -math.sqrt(2.0)) / B
	return tmp

Julia

B = abs(B)
A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	tmp = 0.0
	if (B <= 1.55e+36)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(F * t_0) * Float64(2.0 * A))))) / t_0);
	else
		tmp = Float64(Float64(sqrt(Float64(B * Float64(-F))) * Float64(-sqrt(2.0))) / B);
	end
	return tmp
end

MATLAB

B = abs(B)
A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (A * C));
	tmp = 0.0;
	if (B <= 1.55e+36)
		tmp = -sqrt((2.0 * ((F * t_0) * (2.0 * A)))) / t_0;
	else
		tmp = (sqrt((B * -F)) * -sqrt(2.0)) / B;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, 1.55e+36], N[((-N[Sqrt[N[(2.0 * N[(N[(F * t$95$0), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[N[(B * (-F)), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 1.55e36

   1. Initial program 16.3%

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

   3. Simplified 16.3%

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

   4. Taylor expanded in A around -inf 14.0%

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

      1. *-commutative 14.0%

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

   6. Simplified 14.0%

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

   if 1.55e36 < B

   1. Initial program 8.6%

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

   3. Simplified 8.6%

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

   4. Taylor expanded in C around 0 16.2%

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

      1. mul-1-neg 16.2%

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

      2. *-commutative 16.2%

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

      3. +-commutative 16.2%

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

      4. unpow2 16.2%

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

      5. unpow2 16.2%

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

      6. hypot-def 52.2%

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

   6. Simplified 52.2%

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

      1. associate-*l/ 52.3%

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

   8. Applied egg-rr 52.3%

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

   9. Taylor expanded in A around 0 46.4%

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

       1. mul-1-neg 46.4%

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

       2. distribute-rgt-neg-out 46.4%

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

   11. Simplified 46.4%

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

4. Final simplification 22.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.55 \cdot 10^{+36}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot \left(2 \cdot A\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{B \cdot \left(-F\right)} \cdot \left(-\sqrt{2}\right)}{B}\\ \end{array} \]

Alternative 9: 28.6% accurate, 4.9× speedup

Math

\[\begin{array}{l} B = |B|\\ [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq 4.4 \cdot 10^{+111}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A \cdot F} \cdot \left(-\frac{2}{B}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (+ (* B B) (* -4.0 (* A C)))))
   (if (<= B 4.4e+111)
     (/ (- (sqrt (* 2.0 (* t_0 (* F (* 2.0 A)))))) t_0)
     (* (sqrt (* A F)) (- (/ 2.0 B))))))

C

B = abs(B);
assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) + (-4.0 * (A * C));
	double tmp;
	if (B <= 4.4e+111) {
		tmp = -sqrt((2.0 * (t_0 * (F * (2.0 * A))))) / t_0;
	} else {
		tmp = sqrt((A * F)) * -(2.0 / B);
	}
	return tmp;
}

Fortran

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b * b) + ((-4.0d0) * (a * c))
    if (b <= 4.4d+111) then
        tmp = -sqrt((2.0d0 * (t_0 * (f * (2.0d0 * a))))) / t_0
    else
        tmp = sqrt((a * f)) * -(2.0d0 / b)
    end if
    code = tmp
end function

Java

B = Math.abs(B);
assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) + (-4.0 * (A * C));
	double tmp;
	if (B <= 4.4e+111) {
		tmp = -Math.sqrt((2.0 * (t_0 * (F * (2.0 * A))))) / t_0;
	} else {
		tmp = Math.sqrt((A * F)) * -(2.0 / B);
	}
	return tmp;
}

Python

B = abs(B)
[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (B * B) + (-4.0 * (A * C))
	tmp = 0
	if B <= 4.4e+111:
		tmp = -math.sqrt((2.0 * (t_0 * (F * (2.0 * A))))) / t_0
	else:
		tmp = math.sqrt((A * F)) * -(2.0 / B)
	return tmp

Julia

B = abs(B)
A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) + Float64(-4.0 * Float64(A * C)))
	tmp = 0.0
	if (B <= 4.4e+111)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_0 * Float64(F * Float64(2.0 * A)))))) / t_0);
	else
		tmp = Float64(sqrt(Float64(A * F)) * Float64(-Float64(2.0 / B)));
	end
	return tmp
end

MATLAB

B = abs(B)
A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) + (-4.0 * (A * C));
	tmp = 0.0;
	if (B <= 4.4e+111)
		tmp = -sqrt((2.0 * (t_0 * (F * (2.0 * A))))) / t_0;
	else
		tmp = sqrt((A * F)) * -(2.0 / B);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, 4.4e+111], N[((-N[Sqrt[N[(2.0 * N[(t$95$0 * N[(F * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] * (-N[(2.0 / B), $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 4.39999999999999997e111

   1. Initial program 16.6%

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

   3. Simplified 16.6%

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

   4. Taylor expanded in A around -inf 15.0%

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

      1. *-commutative 15.0%

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

   6. Simplified 15.0%

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

      1. distribute-frac-neg 15.0%

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

      2. associate-*l* 14.5%

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

      3. cancel-sign-sub-inv 14.5%

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

      4. metadata-eval 14.5%

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

      5. cancel-sign-sub-inv 14.5%

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

      6. metadata-eval 14.5%

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

   8. Applied egg-rr 14.5%

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

   if 4.39999999999999997e111 < B

   1. Initial program 4.7%

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

   3. Simplified 4.7%

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

   4. Taylor expanded in C around 0 10.6%

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

      1. mul-1-neg 10.6%

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

      2. *-commutative 10.6%

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

      3. +-commutative 10.6%

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

      4. unpow2 10.6%

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

      5. unpow2 10.6%

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

      6. hypot-def 57.2%

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

   6. Simplified 57.2%

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

   7. Taylor expanded in A around -inf 7.2%

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

      1. count-2 7.2%

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

   9. Simplified 7.2%

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

   10. Taylor expanded in B around 0 7.2%

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

       1. *-commutative 7.2%

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

       2. *-commutative 7.2%

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

       3. unpow2 7.2%

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

       4. rem-square-sqrt 7.2%

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

   12. Simplified 7.2%

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

4. Final simplification 13.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 4.4 \cdot 10^{+111}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A \cdot F} \cdot \left(-\frac{2}{B}\right)\\ \end{array} \]

Alternative 10: 28.6% accurate, 4.9× speedup

Math

\[\begin{array}{l} B = |B|\\ [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq 1.05 \cdot 10^{+108}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(2 \cdot A\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A \cdot F} \cdot \left(-\frac{2}{B}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* A C)))))
   (if (<= B 1.05e+108)
     (/ (- (sqrt (* 2.0 (* (* F t_0) (* 2.0 A))))) t_0)
     (* (sqrt (* A F)) (- (/ 2.0 B))))))

C

B = abs(B);
assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (B <= 1.05e+108) {
		tmp = -sqrt((2.0 * ((F * t_0) * (2.0 * A)))) / t_0;
	} else {
		tmp = sqrt((A * F)) * -(2.0 / B);
	}
	return tmp;
}

Fortran

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b * b) - (4.0d0 * (a * c))
    if (b <= 1.05d+108) then
        tmp = -sqrt((2.0d0 * ((f * t_0) * (2.0d0 * a)))) / t_0
    else
        tmp = sqrt((a * f)) * -(2.0d0 / b)
    end if
    code = tmp
end function

Java

B = Math.abs(B);
assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (B <= 1.05e+108) {
		tmp = -Math.sqrt((2.0 * ((F * t_0) * (2.0 * A)))) / t_0;
	} else {
		tmp = Math.sqrt((A * F)) * -(2.0 / B);
	}
	return tmp;
}

Python

B = abs(B)
[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (A * C))
	tmp = 0
	if B <= 1.05e+108:
		tmp = -math.sqrt((2.0 * ((F * t_0) * (2.0 * A)))) / t_0
	else:
		tmp = math.sqrt((A * F)) * -(2.0 / B)
	return tmp

Julia

B = abs(B)
A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	tmp = 0.0
	if (B <= 1.05e+108)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(F * t_0) * Float64(2.0 * A))))) / t_0);
	else
		tmp = Float64(sqrt(Float64(A * F)) * Float64(-Float64(2.0 / B)));
	end
	return tmp
end

MATLAB

B = abs(B)
A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (A * C));
	tmp = 0.0;
	if (B <= 1.05e+108)
		tmp = -sqrt((2.0 * ((F * t_0) * (2.0 * A)))) / t_0;
	else
		tmp = sqrt((A * F)) * -(2.0 / B);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, 1.05e+108], N[((-N[Sqrt[N[(2.0 * N[(N[(F * t$95$0), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] * (-N[(2.0 / B), $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 1.05000000000000005e108

   1. Initial program 16.6%

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

   3. Simplified 16.6%

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

   4. Taylor expanded in A around -inf 15.0%

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

      1. *-commutative 15.0%

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

   6. Simplified 15.0%

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

   if 1.05000000000000005e108 < B

   1. Initial program 4.7%

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

   3. Simplified 4.7%

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

   4. Taylor expanded in C around 0 10.6%

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

      1. mul-1-neg 10.6%

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

      2. *-commutative 10.6%

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

      3. +-commutative 10.6%

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

      4. unpow2 10.6%

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

      5. unpow2 10.6%

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

      6. hypot-def 57.2%

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

   6. Simplified 57.2%

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

   7. Taylor expanded in A around -inf 7.2%

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

      1. count-2 7.2%

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

   9. Simplified 7.2%

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

   10. Taylor expanded in B around 0 7.2%

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

       1. *-commutative 7.2%

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

       2. *-commutative 7.2%

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

       3. unpow2 7.2%

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

       4. rem-square-sqrt 7.2%

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

   12. Simplified 7.2%

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

4. Final simplification 13.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.05 \cdot 10^{+108}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot \left(2 \cdot A\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A \cdot F} \cdot \left(-\frac{2}{B}\right)\\ \end{array} \]

Alternative 11: 20.9% accurate, 5.0× speedup

Math

\[\begin{array}{l} B = |B|\\ [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -5.8 \cdot 10^{+231}:\\ \;\;\;\;\sqrt{A \cdot F} \cdot \left(-\frac{2}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(2 \cdot A\right) \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (if (<= A -5.8e+231)
   (* (sqrt (* A F)) (- (/ 2.0 B)))
   (/
    (- (sqrt (* 2.0 (* (* 2.0 A) (* -4.0 (* A (* C F)))))))
    (- (* B B) (* 4.0 (* A C))))))

C

B = abs(B);
assert(A < C);
double code(double A, double B, double C, double F) {
	double tmp;
	if (A <= -5.8e+231) {
		tmp = sqrt((A * F)) * -(2.0 / B);
	} else {
		tmp = -sqrt((2.0 * ((2.0 * A) * (-4.0 * (A * (C * F)))))) / ((B * B) - (4.0 * (A * C)));
	}
	return tmp;
}

Fortran

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (a <= (-5.8d+231)) then
        tmp = sqrt((a * f)) * -(2.0d0 / 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

B = Math.abs(B);
assert A < C;
public static double code(double A, double B, double C, double F) {
	double tmp;
	if (A <= -5.8e+231) {
		tmp = Math.sqrt((A * F)) * -(2.0 / B);
	} else {
		tmp = -Math.sqrt((2.0 * ((2.0 * A) * (-4.0 * (A * (C * F)))))) / ((B * B) - (4.0 * (A * C)));
	}
	return tmp;
}

Python

B = abs(B)
[A, C] = sort([A, C])
def code(A, B, C, F):
	tmp = 0
	if A <= -5.8e+231:
		tmp = math.sqrt((A * F)) * -(2.0 / B)
	else:
		tmp = -math.sqrt((2.0 * ((2.0 * A) * (-4.0 * (A * (C * F)))))) / ((B * B) - (4.0 * (A * C)))
	return tmp

Julia

B = abs(B)
A, C = sort([A, C])
function code(A, B, C, F)
	tmp = 0.0
	if (A <= -5.8e+231)
		tmp = Float64(sqrt(Float64(A * F)) * Float64(-Float64(2.0 / 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

B = abs(B)
A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (A <= -5.8e+231)
		tmp = sqrt((A * F)) * -(2.0 / 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: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := If[LessEqual[A, -5.8e+231], N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] * (-N[(2.0 / 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 A < -5.8000000000000002e231

   1. Initial program 1.1%

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

   3. Simplified 1.1%

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

   4. Taylor expanded in C around 0 0.7%

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

      1. mul-1-neg 0.7%

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

      2. *-commutative 0.7%

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

      3. +-commutative 0.7%

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

      4. unpow2 0.7%

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

      5. unpow2 0.7%

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

      6. hypot-def 19.2%

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

   6. Simplified 19.2%

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

   7. Taylor expanded in A around -inf 19.2%

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

      1. count-2 19.2%

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

   9. Simplified 19.2%

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

   10. Taylor expanded in B around 0 19.0%

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

       1. *-commutative 19.0%

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

       2. *-commutative 19.0%

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

       3. unpow2 19.0%

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

       4. rem-square-sqrt 19.3%

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

   12. Simplified 19.3%

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

   if -5.8000000000000002e231 < A

   1. Initial program 15.2%

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

   3. Simplified 15.2%

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

   4. Taylor expanded in A around -inf 12.6%

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

      1. *-commutative 12.6%

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

   6. Simplified 12.6%

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

   7. Taylor expanded in B around 0 11.1%

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

      1. *-commutative 11.1%

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

   9. Simplified 11.1%

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

4. Final simplification 11.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -5.8 \cdot 10^{+231}:\\ \;\;\;\;\sqrt{A \cdot F} \cdot \left(-\frac{2}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(2 \cdot A\right) \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \end{array} \]

Alternative 12: 23.1% accurate, 5.0× speedup

Math

\[\begin{array}{l} B = |B|\\ [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -9 \cdot 10^{+231}:\\ \;\;\;\;\sqrt{A \cdot F} \cdot \left(-\frac{2}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(2 \cdot A\right) \cdot \left(-4 \cdot \left(C \cdot \left(A \cdot F\right)\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (if (<= A -9e+231)
   (* (sqrt (* A F)) (- (/ 2.0 B)))
   (/
    (- (sqrt (* 2.0 (* (* 2.0 A) (* -4.0 (* C (* A F)))))))
    (- (* B B) (* 4.0 (* A C))))))

C

B = abs(B);
assert(A < C);
double code(double A, double B, double C, double F) {
	double tmp;
	if (A <= -9e+231) {
		tmp = sqrt((A * F)) * -(2.0 / B);
	} else {
		tmp = -sqrt((2.0 * ((2.0 * A) * (-4.0 * (C * (A * F)))))) / ((B * B) - (4.0 * (A * C)));
	}
	return tmp;
}

Fortran

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (a <= (-9d+231)) then
        tmp = sqrt((a * f)) * -(2.0d0 / b)
    else
        tmp = -sqrt((2.0d0 * ((2.0d0 * a) * ((-4.0d0) * (c * (a * f)))))) / ((b * b) - (4.0d0 * (a * c)))
    end if
    code = tmp
end function

Java

B = Math.abs(B);
assert A < C;
public static double code(double A, double B, double C, double F) {
	double tmp;
	if (A <= -9e+231) {
		tmp = Math.sqrt((A * F)) * -(2.0 / B);
	} else {
		tmp = -Math.sqrt((2.0 * ((2.0 * A) * (-4.0 * (C * (A * F)))))) / ((B * B) - (4.0 * (A * C)));
	}
	return tmp;
}

Python

B = abs(B)
[A, C] = sort([A, C])
def code(A, B, C, F):
	tmp = 0
	if A <= -9e+231:
		tmp = math.sqrt((A * F)) * -(2.0 / B)
	else:
		tmp = -math.sqrt((2.0 * ((2.0 * A) * (-4.0 * (C * (A * F)))))) / ((B * B) - (4.0 * (A * C)))
	return tmp

Julia

B = abs(B)
A, C = sort([A, C])
function code(A, B, C, F)
	tmp = 0.0
	if (A <= -9e+231)
		tmp = Float64(sqrt(Float64(A * F)) * Float64(-Float64(2.0 / B)));
	else
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(2.0 * A) * Float64(-4.0 * Float64(C * Float64(A * F))))))) / Float64(Float64(B * B) - Float64(4.0 * Float64(A * C))));
	end
	return tmp
end

MATLAB

B = abs(B)
A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (A <= -9e+231)
		tmp = sqrt((A * F)) * -(2.0 / B);
	else
		tmp = -sqrt((2.0 * ((2.0 * A) * (-4.0 * (C * (A * F)))))) / ((B * B) - (4.0 * (A * C)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := If[LessEqual[A, -9e+231], N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] * (-N[(2.0 / B), $MachinePrecision])), $MachinePrecision], N[((-N[Sqrt[N[(2.0 * N[(N[(2.0 * A), $MachinePrecision] * N[(-4.0 * N[(C * N[(A * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -8.99999999999999982e231

   1. Initial program 1.1%

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

   3. Simplified 1.1%

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

   4. Taylor expanded in C around 0 0.7%

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

      1. mul-1-neg 0.7%

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

      2. *-commutative 0.7%

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

      3. +-commutative 0.7%

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

      4. unpow2 0.7%

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

      5. unpow2 0.7%

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

      6. hypot-def 19.2%

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

   6. Simplified 19.2%

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

   7. Taylor expanded in A around -inf 19.2%

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

      1. count-2 19.2%

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

   9. Simplified 19.2%

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

   10. Taylor expanded in B around 0 19.0%

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

       1. *-commutative 19.0%

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

       2. *-commutative 19.0%

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

       3. unpow2 19.0%

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

       4. rem-square-sqrt 19.3%

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

   12. Simplified 19.3%

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

   if -8.99999999999999982e231 < A

   1. Initial program 15.2%

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

   3. Simplified 15.2%

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

   4. Taylor expanded in A around -inf 12.6%

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

      1. *-commutative 12.6%

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

   6. Simplified 12.6%

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

   7. Taylor expanded in B around 0 11.1%

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

      1. *-commutative 11.1%

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

      2. associate-*r* 11.8%

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

      3. *-commutative 11.8%

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

   9. Simplified 11.8%

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

4. Final simplification 12.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -9 \cdot 10^{+231}:\\ \;\;\;\;\sqrt{A \cdot F} \cdot \left(-\frac{2}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(2 \cdot A\right) \cdot \left(-4 \cdot \left(C \cdot \left(A \cdot F\right)\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \end{array} \]

Alternative 13: 19.0% accurate, 5.1× speedup

Math

\[\begin{array}{l} B = |B|\\ [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -1.1 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{A \cdot F} \cdot \left(-\frac{2}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-8 \cdot \left(\left(C \cdot F\right) \cdot \left(A \cdot A\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (if (<= A -1.1e+153)
   (* (sqrt (* A F)) (- (/ 2.0 B)))
   (/
    (- (sqrt (* 2.0 (* -8.0 (* (* C F) (* A A))))))
    (- (* B B) (* 4.0 (* A C))))))

C

B = abs(B);
assert(A < C);
double code(double A, double B, double C, double F) {
	double tmp;
	if (A <= -1.1e+153) {
		tmp = sqrt((A * F)) * -(2.0 / B);
	} else {
		tmp = -sqrt((2.0 * (-8.0 * ((C * F) * (A * A))))) / ((B * B) - (4.0 * (A * C)));
	}
	return tmp;
}

Fortran

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (a <= (-1.1d+153)) then
        tmp = sqrt((a * f)) * -(2.0d0 / b)
    else
        tmp = -sqrt((2.0d0 * ((-8.0d0) * ((c * f) * (a * a))))) / ((b * b) - (4.0d0 * (a * c)))
    end if
    code = tmp
end function

Java

B = Math.abs(B);
assert A < C;
public static double code(double A, double B, double C, double F) {
	double tmp;
	if (A <= -1.1e+153) {
		tmp = Math.sqrt((A * F)) * -(2.0 / B);
	} else {
		tmp = -Math.sqrt((2.0 * (-8.0 * ((C * F) * (A * A))))) / ((B * B) - (4.0 * (A * C)));
	}
	return tmp;
}

Python

B = abs(B)
[A, C] = sort([A, C])
def code(A, B, C, F):
	tmp = 0
	if A <= -1.1e+153:
		tmp = math.sqrt((A * F)) * -(2.0 / B)
	else:
		tmp = -math.sqrt((2.0 * (-8.0 * ((C * F) * (A * A))))) / ((B * B) - (4.0 * (A * C)))
	return tmp

Julia

B = abs(B)
A, C = sort([A, C])
function code(A, B, C, F)
	tmp = 0.0
	if (A <= -1.1e+153)
		tmp = Float64(sqrt(Float64(A * F)) * Float64(-Float64(2.0 / B)));
	else
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(-8.0 * Float64(Float64(C * F) * Float64(A * A)))))) / Float64(Float64(B * B) - Float64(4.0 * Float64(A * C))));
	end
	return tmp
end

MATLAB

B = abs(B)
A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (A <= -1.1e+153)
		tmp = sqrt((A * F)) * -(2.0 / B);
	else
		tmp = -sqrt((2.0 * (-8.0 * ((C * F) * (A * A))))) / ((B * B) - (4.0 * (A * C)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := If[LessEqual[A, -1.1e+153], N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] * (-N[(2.0 / B), $MachinePrecision])), $MachinePrecision], N[((-N[Sqrt[N[(2.0 * N[(-8.0 * N[(N[(C * F), $MachinePrecision] * N[(A * A), $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 A < -1.1e153

   1. Initial program 4.9%

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

   3. Simplified 4.9%

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

   4. Taylor expanded in C around 0 4.5%

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

      1. mul-1-neg 4.5%

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

      2. *-commutative 4.5%

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

      3. +-commutative 4.5%

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

      4. unpow2 4.5%

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

      5. unpow2 4.5%

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

      6. hypot-def 20.3%

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

   6. Simplified 20.3%

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

   7. Taylor expanded in A around -inf 16.5%

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

      1. count-2 16.5%

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

   9. Simplified 16.5%

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

   10. Taylor expanded in B around 0 16.4%

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

       1. *-commutative 16.4%

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

       2. *-commutative 16.4%

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

       3. unpow2 16.4%

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

       4. rem-square-sqrt 16.6%

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

   12. Simplified 16.6%

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

   if -1.1e153 < A

   1. Initial program 15.4%

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

   3. Simplified 15.4%

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

   4. Taylor expanded in A around -inf 12.2%

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

      1. *-commutative 12.2%

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

   6. Simplified 12.2%

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

   7. Taylor expanded in B around 0 10.4%

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

      1. unpow2 10.4%

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

      2. *-commutative 10.4%

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

   9. Simplified 10.4%

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

4. Final simplification 11.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -1.1 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{A \cdot F} \cdot \left(-\frac{2}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-8 \cdot \left(\left(C \cdot F\right) \cdot \left(A \cdot A\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \end{array} \]

Alternative 14: 8.6% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 14.3%

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

3. Simplified 14.3%

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

4. Taylor expanded in A around -inf 12.3%

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

   1. *-commutative 12.3%

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

6. Simplified 12.3%

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

7. Taylor expanded in B around inf 3.7%

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

   1. associate-*r/ 3.7%

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

   2. *-rgt-identity 3.7%

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

   3. *-commutative 3.7%

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

9. Simplified 3.7%

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

10. Final simplification 3.7%

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

Alternative 15: 4.1% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = -((f / a) ** 0.5d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 14.3%

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

3. Simplified 15.8%

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

4. Taylor expanded in A around -inf 8.4%

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

   1. fma-def 8.4%

      \[\leadsto \frac{-\sqrt{F \cdot \left(\left(A - \color{blue}{\mathsf{fma}\left(-0.5, \frac{{B}^{2}}{A}, -1 \cdot A\right)}\right) \cdot \mathsf{fma}\left(C, A \cdot -8, 2 \cdot \left(B \cdot B\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   2. unpow2 8.4%

      \[\leadsto \frac{-\sqrt{F \cdot \left(\left(A - \mathsf{fma}\left(-0.5, \frac{\color{blue}{B \cdot B}}{A}, -1 \cdot A\right)\right) \cdot \mathsf{fma}\left(C, A \cdot -8, 2 \cdot \left(B \cdot B\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   3. mul-1-neg 8.4%

      \[\leadsto \frac{-\sqrt{F \cdot \left(\left(A - \mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, \color{blue}{-A}\right)\right) \cdot \mathsf{fma}\left(C, A \cdot -8, 2 \cdot \left(B \cdot B\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

6. Simplified 8.4%

   \[\leadsto \frac{-\sqrt{F \cdot \left(\left(A - \color{blue}{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, -A\right)}\right) \cdot \mathsf{fma}\left(C, A \cdot -8, 2 \cdot \left(B \cdot B\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

7. Taylor expanded in B around inf 2.5%

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

   1. mul-1-neg 2.5%

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

9. Simplified 2.5%

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

    1. pow1/2 2.8%

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

11. Applied egg-rr 2.8%

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

12. Final simplification 2.8%

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

Alternative 16: 4.0% accurate, 6.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 14.3%

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

3. Simplified 15.8%

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

4. Taylor expanded in A around -inf 8.4%

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

   1. fma-def 8.4%

      \[\leadsto \frac{-\sqrt{F \cdot \left(\left(A - \color{blue}{\mathsf{fma}\left(-0.5, \frac{{B}^{2}}{A}, -1 \cdot A\right)}\right) \cdot \mathsf{fma}\left(C, A \cdot -8, 2 \cdot \left(B \cdot B\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   2. unpow2 8.4%

      \[\leadsto \frac{-\sqrt{F \cdot \left(\left(A - \mathsf{fma}\left(-0.5, \frac{\color{blue}{B \cdot B}}{A}, -1 \cdot A\right)\right) \cdot \mathsf{fma}\left(C, A \cdot -8, 2 \cdot \left(B \cdot B\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   3. mul-1-neg 8.4%

      \[\leadsto \frac{-\sqrt{F \cdot \left(\left(A - \mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, \color{blue}{-A}\right)\right) \cdot \mathsf{fma}\left(C, A \cdot -8, 2 \cdot \left(B \cdot B\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

6. Simplified 8.4%

   \[\leadsto \frac{-\sqrt{F \cdot \left(\left(A - \color{blue}{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, -A\right)}\right) \cdot \mathsf{fma}\left(C, A \cdot -8, 2 \cdot \left(B \cdot B\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

7. Taylor expanded in B around inf 2.5%

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

   1. mul-1-neg 2.5%

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

9. Simplified 2.5%

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

10. Final simplification 2.5%

    \[\leadsto -\sqrt{\frac{F}{A}} \]

Reproduce

herbie shell --seed 2023228 
(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))))