ABCF->ab-angle b

Percentage Accurate: 18.5% → 38.5%

Time: 27.6s

Alternatives: 15

Speedup: 2.9×

• Unsound rule application detected in e-graph. Results may not be sound.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 18.5% 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: 38.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\\ t_1 := {B}^{2} - C \cdot \left(A \cdot 4\right)\\ \mathbf{if}\;B \leq -1.7 \cdot 10^{-57}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(-4 \cdot A\right) \cdot C\right) \cdot t_0\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 1.02 \cdot 10^{+40}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot t_1\right)\right) \cdot \left(A + A\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;{t_0}^{0.5} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (* F (- A (hypot B A)))) (t_1 (- (pow B 2.0) (* C (* A 4.0)))))
   (if (<= B -1.7e-57)
     (/
      (- (sqrt (* 2.0 (* (fma B B (* (* -4.0 A) C)) t_0))))
      (+ (* B B) (* -4.0 (* A C))))
     (if (<= B 1.02e+40)
       (/ (- (sqrt (* (* 2.0 (* F t_1)) (+ A A)))) t_1)
       (* (pow t_0 0.5) (/ (- (sqrt 2.0)) B))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = F * (A - hypot(B, A));
	double t_1 = pow(B, 2.0) - (C * (A * 4.0));
	double tmp;
	if (B <= -1.7e-57) {
		tmp = -sqrt((2.0 * (fma(B, B, ((-4.0 * A) * C)) * t_0))) / ((B * B) + (-4.0 * (A * C)));
	} else if (B <= 1.02e+40) {
		tmp = -sqrt(((2.0 * (F * t_1)) * (A + A))) / t_1;
	} else {
		tmp = pow(t_0, 0.5) * (-sqrt(2.0) / B);
	}
	return tmp;
}

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(F * Float64(A - hypot(B, A)))
	t_1 = Float64((B ^ 2.0) - Float64(C * Float64(A * 4.0)))
	tmp = 0.0
	if (B <= -1.7e-57)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(fma(B, B, Float64(Float64(-4.0 * A) * C)) * t_0)))) / Float64(Float64(B * B) + Float64(-4.0 * Float64(A * C))));
	elseif (B <= 1.02e+40)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(F * t_1)) * Float64(A + A)))) / t_1);
	else
		tmp = Float64((t_0 ^ 0.5) * Float64(Float64(-sqrt(2.0)) / B));
	end
	return tmp
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(F * N[(A - N[Sqrt[B ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[B, 2.0], $MachinePrecision] - N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -1.7e-57], N[((-N[Sqrt[N[(2.0 * N[(N[(B * B + N[(N[(-4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[(B * B), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 1.02e+40], N[((-N[Sqrt[N[(N[(2.0 * N[(F * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(A + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], N[(N[Power[t$95$0, 0.5], $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if B < -1.70000000000000008e-57

   1. Initial program 20.0%

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

   2. Taylor expanded in C around 0 15.9%

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

      1. unpow2 15.9%

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

      2. unpow2 15.9%

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

      3. hypot-def 18.4%

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

   4. Simplified 18.4%

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

      1. *-un-lft-identity 18.4%

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

      2. associate-*l* 18.3%

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

      3. pow2 18.3%

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

      4. associate-*l* 18.3%

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

      5. pow2 18.3%

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

      6. associate-*l* 18.3%

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

   6. Applied egg-rr 18.3%

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

      1. *-lft-identity 18.3%

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

      2. associate-*r* 18.4%

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

      3. unpow2 18.4%

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

      4. cancel-sign-sub-inv 18.4%

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

      5. unpow2 18.4%

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

      6. metadata-eval 18.4%

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

      7. unpow2 18.4%

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

      8. cancel-sign-sub-inv 18.4%

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

   8. Simplified 18.4%

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

      1. *-un-lft-identity 18.4%

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

      2. associate-*l* 18.3%

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

      3. fma-def 18.3%

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

   10. Applied egg-rr 18.3%

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

       1. *-lft-identity 18.3%

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

       2. associate-*l* 19.5%

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

       3. associate-*r* 19.5%

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

   12. Simplified 19.5%

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

   if -1.70000000000000008e-57 < B < 1.02e40

   1. Initial program 21.0%

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

   2. Taylor expanded in C around inf 23.6%

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

      1. sub-neg 23.6%

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

      2. mul-1-neg 23.6%

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

      3. remove-double-neg 23.6%

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

   4. Simplified 23.6%

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

   if 1.02e40 < B

   1. Initial program 11.4%

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

   2. Taylor expanded in C around 0 20.0%

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

      1. mul-1-neg 20.0%

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

      2. *-commutative 20.0%

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

      3. distribute-rgt-neg-in 20.0%

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

      4. *-commutative 20.0%

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

      5. unpow2 20.0%

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

      6. unpow2 20.0%

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

      7. hypot-def 55.4%

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

   4. Simplified 55.4%

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

      1. pow1/2 55.4%

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

   6. Applied egg-rr 55.4%

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

4. Final simplification 28.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -1.7 \cdot 10^{-57}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(-4 \cdot A\right) \cdot C\right) \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 1.02 \cdot 10^{+40}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right)\right) \cdot \left(A + A\right)}}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{0.5} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 2: 38.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := {B}^{2} - C \cdot \left(A \cdot 4\right)\\ t_1 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_2 := A - \mathsf{hypot}\left(B, A\right)\\ \mathbf{if}\;B \leq -1.3 \cdot 10^{-56}:\\ \;\;\;\;\frac{-\sqrt{t_2 \cdot \left(2 \cdot \left(F \cdot t_1\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 3.2 \cdot 10^{+39}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot t_0\right)\right) \cdot \left(A + A\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;{\left(F \cdot t_2\right)}^{0.5} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (pow B 2.0) (* C (* A 4.0))))
        (t_1 (+ (* B B) (* -4.0 (* A C))))
        (t_2 (- A (hypot B A))))
   (if (<= B -1.3e-56)
     (/ (- (sqrt (* t_2 (* 2.0 (* F t_1))))) t_1)
     (if (<= B 3.2e+39)
       (/ (- (sqrt (* (* 2.0 (* F t_0)) (+ A A)))) t_0)
       (* (pow (* F t_2) 0.5) (/ (- (sqrt 2.0)) B))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = pow(B, 2.0) - (C * (A * 4.0));
	double t_1 = (B * B) + (-4.0 * (A * C));
	double t_2 = A - hypot(B, A);
	double tmp;
	if (B <= -1.3e-56) {
		tmp = -sqrt((t_2 * (2.0 * (F * t_1)))) / t_1;
	} else if (B <= 3.2e+39) {
		tmp = -sqrt(((2.0 * (F * t_0)) * (A + A))) / t_0;
	} else {
		tmp = pow((F * t_2), 0.5) * (-sqrt(2.0) / B);
	}
	return tmp;
}

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = Math.pow(B, 2.0) - (C * (A * 4.0));
	double t_1 = (B * B) + (-4.0 * (A * C));
	double t_2 = A - Math.hypot(B, A);
	double tmp;
	if (B <= -1.3e-56) {
		tmp = -Math.sqrt((t_2 * (2.0 * (F * t_1)))) / t_1;
	} else if (B <= 3.2e+39) {
		tmp = -Math.sqrt(((2.0 * (F * t_0)) * (A + A))) / t_0;
	} else {
		tmp = Math.pow((F * t_2), 0.5) * (-Math.sqrt(2.0) / B);
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = math.pow(B, 2.0) - (C * (A * 4.0))
	t_1 = (B * B) + (-4.0 * (A * C))
	t_2 = A - math.hypot(B, A)
	tmp = 0
	if B <= -1.3e-56:
		tmp = -math.sqrt((t_2 * (2.0 * (F * t_1)))) / t_1
	elif B <= 3.2e+39:
		tmp = -math.sqrt(((2.0 * (F * t_0)) * (A + A))) / t_0
	else:
		tmp = math.pow((F * t_2), 0.5) * (-math.sqrt(2.0) / B)
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64((B ^ 2.0) - Float64(C * Float64(A * 4.0)))
	t_1 = Float64(Float64(B * B) + Float64(-4.0 * Float64(A * C)))
	t_2 = Float64(A - hypot(B, A))
	tmp = 0.0
	if (B <= -1.3e-56)
		tmp = Float64(Float64(-sqrt(Float64(t_2 * Float64(2.0 * Float64(F * t_1))))) / t_1);
	elseif (B <= 3.2e+39)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(F * t_0)) * Float64(A + A)))) / t_0);
	else
		tmp = Float64((Float64(F * t_2) ^ 0.5) * Float64(Float64(-sqrt(2.0)) / B));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (B ^ 2.0) - (C * (A * 4.0));
	t_1 = (B * B) + (-4.0 * (A * C));
	t_2 = A - hypot(B, A);
	tmp = 0.0;
	if (B <= -1.3e-56)
		tmp = -sqrt((t_2 * (2.0 * (F * t_1)))) / t_1;
	elseif (B <= 3.2e+39)
		tmp = -sqrt(((2.0 * (F * t_0)) * (A + A))) / t_0;
	else
		tmp = ((F * t_2) ^ 0.5) * (-sqrt(2.0) / B);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[Power[B, 2.0], $MachinePrecision] - N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(B * B), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(A - N[Sqrt[B ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -1.3e-56], N[((-N[Sqrt[N[(t$95$2 * N[(2.0 * N[(F * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], If[LessEqual[B, 3.2e+39], N[((-N[Sqrt[N[(N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(A + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(N[Power[N[(F * t$95$2), $MachinePrecision], 0.5], $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if B < -1.29999999999999998e-56

   1. Initial program 20.0%

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

   2. Taylor expanded in C around 0 15.9%

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

      1. unpow2 15.9%

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

      2. unpow2 15.9%

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

      3. hypot-def 18.4%

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

   4. Simplified 18.4%

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

      1. *-un-lft-identity 18.4%

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

      2. associate-*l* 18.3%

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

      3. pow2 18.3%

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

      4. associate-*l* 18.3%

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

      5. pow2 18.3%

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

      6. associate-*l* 18.3%

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

   6. Applied egg-rr 18.3%

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

      1. *-lft-identity 18.3%

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

      2. associate-*r* 18.4%

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

      3. unpow2 18.4%

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

      4. cancel-sign-sub-inv 18.4%

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

      5. unpow2 18.4%

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

      6. metadata-eval 18.4%

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

      7. unpow2 18.4%

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

      8. cancel-sign-sub-inv 18.4%

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

   8. Simplified 18.4%

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

   if -1.29999999999999998e-56 < B < 3.19999999999999993e39

   1. Initial program 21.0%

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

   2. Taylor expanded in C around inf 23.6%

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

      1. sub-neg 23.6%

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

      2. mul-1-neg 23.6%

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

      3. remove-double-neg 23.6%

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

   4. Simplified 23.6%

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

   if 3.19999999999999993e39 < B

   1. Initial program 11.4%

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

   2. Taylor expanded in C around 0 20.0%

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

      1. mul-1-neg 20.0%

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

      2. *-commutative 20.0%

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

      3. distribute-rgt-neg-in 20.0%

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

      4. *-commutative 20.0%

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

      5. unpow2 20.0%

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

      6. unpow2 20.0%

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

      7. hypot-def 55.4%

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

   4. Simplified 55.4%

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

      1. pow1/2 55.4%

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

   6. Applied egg-rr 55.4%

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

4. Final simplification 27.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -1.3 \cdot 10^{-56}:\\ \;\;\;\;\frac{-\sqrt{\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot \left(2 \cdot \left(F \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 3.2 \cdot 10^{+39}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right)\right) \cdot \left(A + A\right)}}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{0.5} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 3: 34.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := A - \mathsf{hypot}\left(B, A\right)\\ t_2 := \frac{-\sqrt{t_1 \cdot \left(2 \cdot \left(F \cdot t_0\right)\right)}}{t_0}\\ \mathbf{if}\;B \leq -3.7 \cdot 10^{-203}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 1.8 \cdot 10^{-237}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(\left(A \cdot A\right) \cdot \left(C \cdot F\right)\right)}}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;B \leq 1.35 \cdot 10^{+41}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;{\left(F \cdot t_1\right)}^{0.5} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (+ (* B B) (* -4.0 (* A C))))
        (t_1 (- A (hypot B A)))
        (t_2 (/ (- (sqrt (* t_1 (* 2.0 (* F t_0))))) t_0)))
   (if (<= B -3.7e-203)
     t_2
     (if (<= B 1.8e-237)
       (/
        (- (sqrt (* -16.0 (* (* A A) (* C F)))))
        (- (pow B 2.0) (* C (* A 4.0))))
       (if (<= B 1.35e+41)
         t_2
         (* (pow (* F t_1) 0.5) (/ (- (sqrt 2.0)) B)))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) + (-4.0 * (A * C));
	double t_1 = A - hypot(B, A);
	double t_2 = -sqrt((t_1 * (2.0 * (F * t_0)))) / t_0;
	double tmp;
	if (B <= -3.7e-203) {
		tmp = t_2;
	} else if (B <= 1.8e-237) {
		tmp = -sqrt((-16.0 * ((A * A) * (C * F)))) / (pow(B, 2.0) - (C * (A * 4.0)));
	} else if (B <= 1.35e+41) {
		tmp = t_2;
	} else {
		tmp = pow((F * t_1), 0.5) * (-sqrt(2.0) / B);
	}
	return tmp;
}

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) + (-4.0 * (A * C));
	double t_1 = A - Math.hypot(B, A);
	double t_2 = -Math.sqrt((t_1 * (2.0 * (F * t_0)))) / t_0;
	double tmp;
	if (B <= -3.7e-203) {
		tmp = t_2;
	} else if (B <= 1.8e-237) {
		tmp = -Math.sqrt((-16.0 * ((A * A) * (C * F)))) / (Math.pow(B, 2.0) - (C * (A * 4.0)));
	} else if (B <= 1.35e+41) {
		tmp = t_2;
	} else {
		tmp = Math.pow((F * t_1), 0.5) * (-Math.sqrt(2.0) / B);
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (B * B) + (-4.0 * (A * C))
	t_1 = A - math.hypot(B, A)
	t_2 = -math.sqrt((t_1 * (2.0 * (F * t_0)))) / t_0
	tmp = 0
	if B <= -3.7e-203:
		tmp = t_2
	elif B <= 1.8e-237:
		tmp = -math.sqrt((-16.0 * ((A * A) * (C * F)))) / (math.pow(B, 2.0) - (C * (A * 4.0)))
	elif B <= 1.35e+41:
		tmp = t_2
	else:
		tmp = math.pow((F * t_1), 0.5) * (-math.sqrt(2.0) / B)
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) + Float64(-4.0 * Float64(A * C)))
	t_1 = Float64(A - hypot(B, A))
	t_2 = Float64(Float64(-sqrt(Float64(t_1 * Float64(2.0 * Float64(F * t_0))))) / t_0)
	tmp = 0.0
	if (B <= -3.7e-203)
		tmp = t_2;
	elseif (B <= 1.8e-237)
		tmp = Float64(Float64(-sqrt(Float64(-16.0 * Float64(Float64(A * A) * Float64(C * F))))) / Float64((B ^ 2.0) - Float64(C * Float64(A * 4.0))));
	elseif (B <= 1.35e+41)
		tmp = t_2;
	else
		tmp = Float64((Float64(F * t_1) ^ 0.5) * Float64(Float64(-sqrt(2.0)) / B));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) + (-4.0 * (A * C));
	t_1 = A - hypot(B, A);
	t_2 = -sqrt((t_1 * (2.0 * (F * t_0)))) / t_0;
	tmp = 0.0;
	if (B <= -3.7e-203)
		tmp = t_2;
	elseif (B <= 1.8e-237)
		tmp = -sqrt((-16.0 * ((A * A) * (C * F)))) / ((B ^ 2.0) - (C * (A * 4.0)));
	elseif (B <= 1.35e+41)
		tmp = t_2;
	else
		tmp = ((F * t_1) ^ 0.5) * (-sqrt(2.0) / B);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(A - N[Sqrt[B ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-N[Sqrt[N[(t$95$1 * N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision]}, If[LessEqual[B, -3.7e-203], t$95$2, If[LessEqual[B, 1.8e-237], N[((-N[Sqrt[N[(-16.0 * N[(N[(A * A), $MachinePrecision] * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[Power[B, 2.0], $MachinePrecision] - N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 1.35e+41], t$95$2, N[(N[Power[N[(F * t$95$1), $MachinePrecision], 0.5], $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if B < -3.70000000000000002e-203 or 1.79999999999999998e-237 < B < 1.35e41

   1. Initial program 22.0%

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

   2. Taylor expanded in C around 0 18.3%

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

      1. unpow2 18.3%

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

      2. unpow2 18.3%

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

      3. hypot-def 21.3%

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

   4. Simplified 21.3%

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

      1. *-un-lft-identity 21.3%

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

      2. associate-*l* 21.3%

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

      3. pow2 21.3%

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

      4. associate-*l* 21.3%

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

      5. pow2 21.3%

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

      6. associate-*l* 21.3%

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

   6. Applied egg-rr 21.3%

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

      1. *-lft-identity 21.3%

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

      2. associate-*r* 21.3%

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

      3. unpow2 21.3%

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

      4. cancel-sign-sub-inv 21.3%

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

      5. unpow2 21.3%

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

      6. metadata-eval 21.3%

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

      7. unpow2 21.3%

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

      8. cancel-sign-sub-inv 21.3%

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

   8. Simplified 21.3%

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

   if -3.70000000000000002e-203 < B < 1.79999999999999998e-237

   1. Initial program 12.5%

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

   2. Taylor expanded in C around 0 15.4%

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

      1. unpow2 15.4%

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

      2. unpow2 15.4%

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

      3. hypot-def 16.1%

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

   4. Simplified 16.1%

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

   5. Taylor expanded in A around -inf 22.2%

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

      1. unpow2 22.2%

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

   7. Simplified 22.2%

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

   if 1.35e41 < B

   1. Initial program 11.4%

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

   2. Taylor expanded in C around 0 20.0%

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

      1. mul-1-neg 20.0%

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

      2. *-commutative 20.0%

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

      3. distribute-rgt-neg-in 20.0%

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

      4. *-commutative 20.0%

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

      5. unpow2 20.0%

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

      6. unpow2 20.0%

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

      7. hypot-def 55.4%

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

   4. Simplified 55.4%

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

      1. pow1/2 55.4%

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

   6. Applied egg-rr 55.4%

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

4. Final simplification 27.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -3.7 \cdot 10^{-203}:\\ \;\;\;\;\frac{-\sqrt{\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot \left(2 \cdot \left(F \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 1.8 \cdot 10^{-237}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(\left(A \cdot A\right) \cdot \left(C \cdot F\right)\right)}}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;B \leq 1.35 \cdot 10^{+41}:\\ \;\;\;\;\frac{-\sqrt{\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot \left(2 \cdot \left(F \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{0.5} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 4: 34.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := A - \mathsf{hypot}\left(B, A\right)\\ t_2 := \frac{-\sqrt{t_1 \cdot \left(2 \cdot \left(F \cdot t_0\right)\right)}}{t_0}\\ \mathbf{if}\;B \leq -4.8 \cdot 10^{-203}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 3.8 \cdot 10^{-242}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(\left(A \cdot A\right) \cdot \left(C \cdot F\right)\right)}}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;B \leq 5.5 \cdot 10^{+40}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot t_1}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (+ (* B B) (* -4.0 (* A C))))
        (t_1 (- A (hypot B A)))
        (t_2 (/ (- (sqrt (* t_1 (* 2.0 (* F t_0))))) t_0)))
   (if (<= B -4.8e-203)
     t_2
     (if (<= B 3.8e-242)
       (/
        (- (sqrt (* -16.0 (* (* A A) (* C F)))))
        (- (pow B 2.0) (* C (* A 4.0))))
       (if (<= B 5.5e+40) t_2 (* (/ (sqrt 2.0) B) (- (sqrt (* F t_1)))))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) + (-4.0 * (A * C));
	double t_1 = A - hypot(B, A);
	double t_2 = -sqrt((t_1 * (2.0 * (F * t_0)))) / t_0;
	double tmp;
	if (B <= -4.8e-203) {
		tmp = t_2;
	} else if (B <= 3.8e-242) {
		tmp = -sqrt((-16.0 * ((A * A) * (C * F)))) / (pow(B, 2.0) - (C * (A * 4.0)));
	} else if (B <= 5.5e+40) {
		tmp = t_2;
	} else {
		tmp = (sqrt(2.0) / B) * -sqrt((F * t_1));
	}
	return tmp;
}

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) + (-4.0 * (A * C));
	double t_1 = A - Math.hypot(B, A);
	double t_2 = -Math.sqrt((t_1 * (2.0 * (F * t_0)))) / t_0;
	double tmp;
	if (B <= -4.8e-203) {
		tmp = t_2;
	} else if (B <= 3.8e-242) {
		tmp = -Math.sqrt((-16.0 * ((A * A) * (C * F)))) / (Math.pow(B, 2.0) - (C * (A * 4.0)));
	} else if (B <= 5.5e+40) {
		tmp = t_2;
	} else {
		tmp = (Math.sqrt(2.0) / B) * -Math.sqrt((F * t_1));
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (B * B) + (-4.0 * (A * C))
	t_1 = A - math.hypot(B, A)
	t_2 = -math.sqrt((t_1 * (2.0 * (F * t_0)))) / t_0
	tmp = 0
	if B <= -4.8e-203:
		tmp = t_2
	elif B <= 3.8e-242:
		tmp = -math.sqrt((-16.0 * ((A * A) * (C * F)))) / (math.pow(B, 2.0) - (C * (A * 4.0)))
	elif B <= 5.5e+40:
		tmp = t_2
	else:
		tmp = (math.sqrt(2.0) / B) * -math.sqrt((F * t_1))
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) + Float64(-4.0 * Float64(A * C)))
	t_1 = Float64(A - hypot(B, A))
	t_2 = Float64(Float64(-sqrt(Float64(t_1 * Float64(2.0 * Float64(F * t_0))))) / t_0)
	tmp = 0.0
	if (B <= -4.8e-203)
		tmp = t_2;
	elseif (B <= 3.8e-242)
		tmp = Float64(Float64(-sqrt(Float64(-16.0 * Float64(Float64(A * A) * Float64(C * F))))) / Float64((B ^ 2.0) - Float64(C * Float64(A * 4.0))));
	elseif (B <= 5.5e+40)
		tmp = t_2;
	else
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(-sqrt(Float64(F * t_1))));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) + (-4.0 * (A * C));
	t_1 = A - hypot(B, A);
	t_2 = -sqrt((t_1 * (2.0 * (F * t_0)))) / t_0;
	tmp = 0.0;
	if (B <= -4.8e-203)
		tmp = t_2;
	elseif (B <= 3.8e-242)
		tmp = -sqrt((-16.0 * ((A * A) * (C * F)))) / ((B ^ 2.0) - (C * (A * 4.0)));
	elseif (B <= 5.5e+40)
		tmp = t_2;
	else
		tmp = (sqrt(2.0) / B) * -sqrt((F * t_1));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(A - N[Sqrt[B ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-N[Sqrt[N[(t$95$1 * N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision]}, If[LessEqual[B, -4.8e-203], t$95$2, If[LessEqual[B, 3.8e-242], N[((-N[Sqrt[N[(-16.0 * N[(N[(A * A), $MachinePrecision] * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[Power[B, 2.0], $MachinePrecision] - N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 5.5e+40], t$95$2, N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * (-N[Sqrt[N[(F * t$95$1), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if B < -4.7999999999999997e-203 or 3.8000000000000002e-242 < B < 5.49999999999999974e40

   1. Initial program 22.0%

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

   2. Taylor expanded in C around 0 18.3%

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

      1. unpow2 18.3%

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

      2. unpow2 18.3%

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

      3. hypot-def 21.3%

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

   4. Simplified 21.3%

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

      1. *-un-lft-identity 21.3%

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

      2. associate-*l* 21.3%

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

      3. pow2 21.3%

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

      4. associate-*l* 21.3%

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

      5. pow2 21.3%

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

      6. associate-*l* 21.3%

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

   6. Applied egg-rr 21.3%

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

      1. *-lft-identity 21.3%

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

      2. associate-*r* 21.3%

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

      3. unpow2 21.3%

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

      4. cancel-sign-sub-inv 21.3%

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

      5. unpow2 21.3%

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

      6. metadata-eval 21.3%

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

      7. unpow2 21.3%

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

      8. cancel-sign-sub-inv 21.3%

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

   8. Simplified 21.3%

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

   if -4.7999999999999997e-203 < B < 3.8000000000000002e-242

   1. Initial program 12.5%

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

   2. Taylor expanded in C around 0 15.4%

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

      1. unpow2 15.4%

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

      2. unpow2 15.4%

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

      3. hypot-def 16.1%

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

   4. Simplified 16.1%

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

   5. Taylor expanded in A around -inf 22.2%

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

      1. unpow2 22.2%

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

   7. Simplified 22.2%

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

   if 5.49999999999999974e40 < B

   1. Initial program 11.4%

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

   2. Taylor expanded in C around 0 20.0%

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

      1. mul-1-neg 20.0%

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

      2. *-commutative 20.0%

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

      3. distribute-rgt-neg-in 20.0%

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

      4. *-commutative 20.0%

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

      5. unpow2 20.0%

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

      6. unpow2 20.0%

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

      7. hypot-def 55.4%

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

   4. Simplified 55.4%

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

4. Final simplification 27.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -4.8 \cdot 10^{-203}:\\ \;\;\;\;\frac{-\sqrt{\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot \left(2 \cdot \left(F \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 3.8 \cdot 10^{-242}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(\left(A \cdot A\right) \cdot \left(C \cdot F\right)\right)}}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;B \leq 5.5 \cdot 10^{+40}:\\ \;\;\;\;\frac{-\sqrt{\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot \left(2 \cdot \left(F \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\right)\\ \end{array} \]

Alternative 5: 32.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := \sqrt{\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot \left(2 \cdot \left(F \cdot t_0\right)\right)}\\ \mathbf{if}\;B \leq -5.1 \cdot 10^{-203}:\\ \;\;\;\;\frac{-t_1}{t_0}\\ \mathbf{elif}\;B \leq 1.5 \cdot 10^{-233}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(\left(A \cdot A\right) \cdot \left(C \cdot F\right)\right)}}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;B \leq 1.05 \cdot 10^{+45}:\\ \;\;\;\;t_1 \cdot \frac{-1}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - B\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (+ (* B B) (* -4.0 (* A C))))
        (t_1 (sqrt (* (- A (hypot B A)) (* 2.0 (* F t_0))))))
   (if (<= B -5.1e-203)
     (/ (- t_1) t_0)
     (if (<= B 1.5e-233)
       (/
        (- (sqrt (* -16.0 (* (* A A) (* C F)))))
        (- (pow B 2.0) (* C (* A 4.0))))
       (if (<= B 1.05e+45)
         (* t_1 (/ -1.0 t_0))
         (* (/ (- (sqrt 2.0)) B) (sqrt (* F (- A B)))))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) + (-4.0 * (A * C));
	double t_1 = sqrt(((A - hypot(B, A)) * (2.0 * (F * t_0))));
	double tmp;
	if (B <= -5.1e-203) {
		tmp = -t_1 / t_0;
	} else if (B <= 1.5e-233) {
		tmp = -sqrt((-16.0 * ((A * A) * (C * F)))) / (pow(B, 2.0) - (C * (A * 4.0)));
	} else if (B <= 1.05e+45) {
		tmp = t_1 * (-1.0 / t_0);
	} else {
		tmp = (-sqrt(2.0) / B) * sqrt((F * (A - B)));
	}
	return tmp;
}

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) + (-4.0 * (A * C));
	double t_1 = Math.sqrt(((A - Math.hypot(B, A)) * (2.0 * (F * t_0))));
	double tmp;
	if (B <= -5.1e-203) {
		tmp = -t_1 / t_0;
	} else if (B <= 1.5e-233) {
		tmp = -Math.sqrt((-16.0 * ((A * A) * (C * F)))) / (Math.pow(B, 2.0) - (C * (A * 4.0)));
	} else if (B <= 1.05e+45) {
		tmp = t_1 * (-1.0 / t_0);
	} else {
		tmp = (-Math.sqrt(2.0) / B) * Math.sqrt((F * (A - B)));
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (B * B) + (-4.0 * (A * C))
	t_1 = math.sqrt(((A - math.hypot(B, A)) * (2.0 * (F * t_0))))
	tmp = 0
	if B <= -5.1e-203:
		tmp = -t_1 / t_0
	elif B <= 1.5e-233:
		tmp = -math.sqrt((-16.0 * ((A * A) * (C * F)))) / (math.pow(B, 2.0) - (C * (A * 4.0)))
	elif B <= 1.05e+45:
		tmp = t_1 * (-1.0 / t_0)
	else:
		tmp = (-math.sqrt(2.0) / B) * math.sqrt((F * (A - B)))
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) + Float64(-4.0 * Float64(A * C)))
	t_1 = sqrt(Float64(Float64(A - hypot(B, A)) * Float64(2.0 * Float64(F * t_0))))
	tmp = 0.0
	if (B <= -5.1e-203)
		tmp = Float64(Float64(-t_1) / t_0);
	elseif (B <= 1.5e-233)
		tmp = Float64(Float64(-sqrt(Float64(-16.0 * Float64(Float64(A * A) * Float64(C * F))))) / Float64((B ^ 2.0) - Float64(C * Float64(A * 4.0))));
	elseif (B <= 1.05e+45)
		tmp = Float64(t_1 * Float64(-1.0 / t_0));
	else
		tmp = Float64(Float64(Float64(-sqrt(2.0)) / B) * sqrt(Float64(F * Float64(A - B))));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) + (-4.0 * (A * C));
	t_1 = sqrt(((A - hypot(B, A)) * (2.0 * (F * t_0))));
	tmp = 0.0;
	if (B <= -5.1e-203)
		tmp = -t_1 / t_0;
	elseif (B <= 1.5e-233)
		tmp = -sqrt((-16.0 * ((A * A) * (C * F)))) / ((B ^ 2.0) - (C * (A * 4.0)));
	elseif (B <= 1.05e+45)
		tmp = t_1 * (-1.0 / t_0);
	else
		tmp = (-sqrt(2.0) / B) * sqrt((F * (A - B)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(A - N[Sqrt[B ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[B, -5.1e-203], N[((-t$95$1) / t$95$0), $MachinePrecision], If[LessEqual[B, 1.5e-233], N[((-N[Sqrt[N[(-16.0 * N[(N[(A * A), $MachinePrecision] * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[Power[B, 2.0], $MachinePrecision] - N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 1.05e+45], N[(t$95$1 * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[((-N[Sqrt[2.0], $MachinePrecision]) / B), $MachinePrecision] * N[Sqrt[N[(F * N[(A - B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < -5.09999999999999985e-203

   1. Initial program 22.1%

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

   2. Taylor expanded in C around 0 17.7%

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

      1. unpow2 17.7%

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

      2. unpow2 17.7%

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

      3. hypot-def 19.9%

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

   4. Simplified 19.9%

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

      1. *-un-lft-identity 19.9%

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

      2. associate-*l* 19.9%

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

      3. pow2 19.9%

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

      4. associate-*l* 19.9%

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

      5. pow2 19.9%

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

      6. associate-*l* 19.9%

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

   6. Applied egg-rr 19.9%

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

      1. *-lft-identity 19.9%

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

      2. associate-*r* 19.9%

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

      3. unpow2 19.9%

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

      4. cancel-sign-sub-inv 19.9%

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

      5. unpow2 19.9%

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

      6. metadata-eval 19.9%

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

      7. unpow2 19.9%

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

      8. cancel-sign-sub-inv 19.9%

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

   8. Simplified 19.9%

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

   if -5.09999999999999985e-203 < B < 1.49999999999999999e-233

   1. Initial program 12.5%

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

   2. Taylor expanded in C around 0 15.4%

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

      1. unpow2 15.4%

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

      2. unpow2 15.4%

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

      3. hypot-def 16.1%

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

   4. Simplified 16.1%

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

   5. Taylor expanded in A around -inf 22.2%

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

      1. unpow2 22.2%

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

   7. Simplified 22.2%

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

   if 1.49999999999999999e-233 < B < 1.04999999999999997e45

   1. Initial program 21.5%

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

   2. Taylor expanded in C around 0 19.1%

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

      1. unpow2 19.1%

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

      2. unpow2 19.1%

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

      3. hypot-def 23.4%

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

   4. Simplified 23.4%

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

      1. div-inv 23.4%

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

      2. associate-*l* 23.4%

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

      3. pow2 23.4%

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

      4. associate-*l* 23.4%

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

      5. pow2 23.4%

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

      6. associate-*l* 23.4%

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

   6. Applied egg-rr 23.4%

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

      1. associate-*r* 23.4%

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

      2. unpow2 23.4%

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

      3. cancel-sign-sub-inv 23.4%

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

      4. unpow2 23.4%

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

      5. metadata-eval 23.4%

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

      6. unpow2 23.4%

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

      7. cancel-sign-sub-inv 23.4%

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

      8. unpow2 23.4%

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

      9. metadata-eval 23.4%

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

   8. Simplified 23.4%

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

   if 1.04999999999999997e45 < B

   1. Initial program 11.6%

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

   2. Taylor expanded in B around inf 8.7%

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

   3. Taylor expanded in C around 0 56.5%

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

      1. associate-*r* 56.5%

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

      2. neg-mul-1 56.5%

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

      3. distribute-neg-frac 56.5%

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

   5. Simplified 56.5%

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

4. Final simplification 27.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -5.1 \cdot 10^{-203}:\\ \;\;\;\;\frac{-\sqrt{\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot \left(2 \cdot \left(F \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 1.5 \cdot 10^{-233}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(\left(A \cdot A\right) \cdot \left(C \cdot F\right)\right)}}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;B \leq 1.05 \cdot 10^{+45}:\\ \;\;\;\;\sqrt{\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot \left(2 \cdot \left(F \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right)\right)} \cdot \frac{-1}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - B\right)}\\ \end{array} \]

Alternative 6: 32.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := \frac{-\sqrt{\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot \left(2 \cdot \left(F \cdot t_0\right)\right)}}{t_0}\\ \mathbf{if}\;B \leq -2.95 \cdot 10^{-203}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 6.5 \cdot 10^{-236}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(\left(A \cdot A\right) \cdot \left(C \cdot F\right)\right)}}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;B \leq 2.3 \cdot 10^{+45}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - B\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (+ (* B B) (* -4.0 (* A C))))
        (t_1 (/ (- (sqrt (* (- A (hypot B A)) (* 2.0 (* F t_0))))) t_0)))
   (if (<= B -2.95e-203)
     t_1
     (if (<= B 6.5e-236)
       (/
        (- (sqrt (* -16.0 (* (* A A) (* C F)))))
        (- (pow B 2.0) (* C (* A 4.0))))
       (if (<= B 2.3e+45)
         t_1
         (* (/ (- (sqrt 2.0)) B) (sqrt (* F (- A B)))))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) + (-4.0 * (A * C));
	double t_1 = -sqrt(((A - hypot(B, A)) * (2.0 * (F * t_0)))) / t_0;
	double tmp;
	if (B <= -2.95e-203) {
		tmp = t_1;
	} else if (B <= 6.5e-236) {
		tmp = -sqrt((-16.0 * ((A * A) * (C * F)))) / (pow(B, 2.0) - (C * (A * 4.0)));
	} else if (B <= 2.3e+45) {
		tmp = t_1;
	} else {
		tmp = (-sqrt(2.0) / B) * sqrt((F * (A - B)));
	}
	return tmp;
}

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) + (-4.0 * (A * C));
	double t_1 = -Math.sqrt(((A - Math.hypot(B, A)) * (2.0 * (F * t_0)))) / t_0;
	double tmp;
	if (B <= -2.95e-203) {
		tmp = t_1;
	} else if (B <= 6.5e-236) {
		tmp = -Math.sqrt((-16.0 * ((A * A) * (C * F)))) / (Math.pow(B, 2.0) - (C * (A * 4.0)));
	} else if (B <= 2.3e+45) {
		tmp = t_1;
	} else {
		tmp = (-Math.sqrt(2.0) / B) * Math.sqrt((F * (A - B)));
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (B * B) + (-4.0 * (A * C))
	t_1 = -math.sqrt(((A - math.hypot(B, A)) * (2.0 * (F * t_0)))) / t_0
	tmp = 0
	if B <= -2.95e-203:
		tmp = t_1
	elif B <= 6.5e-236:
		tmp = -math.sqrt((-16.0 * ((A * A) * (C * F)))) / (math.pow(B, 2.0) - (C * (A * 4.0)))
	elif B <= 2.3e+45:
		tmp = t_1
	else:
		tmp = (-math.sqrt(2.0) / B) * math.sqrt((F * (A - B)))
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) + Float64(-4.0 * Float64(A * C)))
	t_1 = Float64(Float64(-sqrt(Float64(Float64(A - hypot(B, A)) * Float64(2.0 * Float64(F * t_0))))) / t_0)
	tmp = 0.0
	if (B <= -2.95e-203)
		tmp = t_1;
	elseif (B <= 6.5e-236)
		tmp = Float64(Float64(-sqrt(Float64(-16.0 * Float64(Float64(A * A) * Float64(C * F))))) / Float64((B ^ 2.0) - Float64(C * Float64(A * 4.0))));
	elseif (B <= 2.3e+45)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(-sqrt(2.0)) / B) * sqrt(Float64(F * Float64(A - B))));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) + (-4.0 * (A * C));
	t_1 = -sqrt(((A - hypot(B, A)) * (2.0 * (F * t_0)))) / t_0;
	tmp = 0.0;
	if (B <= -2.95e-203)
		tmp = t_1;
	elseif (B <= 6.5e-236)
		tmp = -sqrt((-16.0 * ((A * A) * (C * F)))) / ((B ^ 2.0) - (C * (A * 4.0)));
	elseif (B <= 2.3e+45)
		tmp = t_1;
	else
		tmp = (-sqrt(2.0) / B) * sqrt((F * (A - B)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-N[Sqrt[N[(N[(A - N[Sqrt[B ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision]}, If[LessEqual[B, -2.95e-203], t$95$1, If[LessEqual[B, 6.5e-236], N[((-N[Sqrt[N[(-16.0 * N[(N[(A * A), $MachinePrecision] * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[Power[B, 2.0], $MachinePrecision] - N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 2.3e+45], t$95$1, N[(N[((-N[Sqrt[2.0], $MachinePrecision]) / B), $MachinePrecision] * N[Sqrt[N[(F * N[(A - B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if B < -2.95e-203 or 6.5000000000000001e-236 < B < 2.30000000000000012e45

   1. Initial program 21.9%

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

   2. Taylor expanded in C around 0 18.2%

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

      1. unpow2 18.2%

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

      2. unpow2 18.2%

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

      3. hypot-def 21.2%

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

   4. Simplified 21.2%

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

      1. *-un-lft-identity 21.2%

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

      2. associate-*l* 21.2%

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

      3. pow2 21.2%

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

      4. associate-*l* 21.2%

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

      5. pow2 21.2%

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

      6. associate-*l* 21.2%

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

   6. Applied egg-rr 21.2%

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

      1. *-lft-identity 21.2%

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

      2. associate-*r* 21.2%

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

      3. unpow2 21.2%

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

      4. cancel-sign-sub-inv 21.2%

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

      5. unpow2 21.2%

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

      6. metadata-eval 21.2%

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

      7. unpow2 21.2%

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

      8. cancel-sign-sub-inv 21.2%

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

   8. Simplified 21.2%

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

   if -2.95e-203 < B < 6.5000000000000001e-236

   1. Initial program 12.5%

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

   2. Taylor expanded in C around 0 15.4%

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

      1. unpow2 15.4%

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

      2. unpow2 15.4%

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

      3. hypot-def 16.1%

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

   4. Simplified 16.1%

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

   5. Taylor expanded in A around -inf 22.2%

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

      1. unpow2 22.2%

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

   7. Simplified 22.2%

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

   if 2.30000000000000012e45 < B

   1. Initial program 11.6%

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

   2. Taylor expanded in B around inf 8.7%

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

   3. Taylor expanded in C around 0 56.5%

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

      1. associate-*r* 56.5%

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

      2. neg-mul-1 56.5%

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

      3. distribute-neg-frac 56.5%

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

   5. Simplified 56.5%

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

4. Final simplification 27.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -2.95 \cdot 10^{-203}:\\ \;\;\;\;\frac{-\sqrt{\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot \left(2 \cdot \left(F \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 6.5 \cdot 10^{-236}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(\left(A \cdot A\right) \cdot \left(C \cdot F\right)\right)}}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;B \leq 2.3 \cdot 10^{+45}:\\ \;\;\;\;\frac{-\sqrt{\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot \left(2 \cdot \left(F \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - B\right)}\\ \end{array} \]

Alternative 7: 27.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := A - \mathsf{hypot}\left(B, A\right)\\ \mathbf{if}\;B \leq -1.55 \cdot 10^{+28}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(2 \cdot \left(F \cdot \left(B \cdot B\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 4.5 \cdot 10^{-242}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(\left(A \cdot A\right) \cdot \left(C \cdot F\right)\right)}}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;B \leq 4.7 \cdot 10^{-83}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(2 \cdot \left(\left(\left(-4 \cdot A\right) \cdot C\right) \cdot F\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - B\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (+ (* B B) (* -4.0 (* A C)))) (t_1 (- A (hypot B A))))
   (if (<= B -1.55e+28)
     (/ (- (sqrt (* t_1 (* 2.0 (* F (* B B)))))) t_0)
     (if (<= B 4.5e-242)
       (/
        (- (sqrt (* -16.0 (* (* A A) (* C F)))))
        (- (pow B 2.0) (* C (* A 4.0))))
       (if (<= B 4.7e-83)
         (/ (- (sqrt (* t_1 (* 2.0 (* (* (* -4.0 A) C) F))))) t_0)
         (* (/ (- (sqrt 2.0)) B) (sqrt (* F (- A B)))))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) + (-4.0 * (A * C));
	double t_1 = A - hypot(B, A);
	double tmp;
	if (B <= -1.55e+28) {
		tmp = -sqrt((t_1 * (2.0 * (F * (B * B))))) / t_0;
	} else if (B <= 4.5e-242) {
		tmp = -sqrt((-16.0 * ((A * A) * (C * F)))) / (pow(B, 2.0) - (C * (A * 4.0)));
	} else if (B <= 4.7e-83) {
		tmp = -sqrt((t_1 * (2.0 * (((-4.0 * A) * C) * F)))) / t_0;
	} else {
		tmp = (-sqrt(2.0) / B) * sqrt((F * (A - B)));
	}
	return tmp;
}

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) + (-4.0 * (A * C));
	double t_1 = A - Math.hypot(B, A);
	double tmp;
	if (B <= -1.55e+28) {
		tmp = -Math.sqrt((t_1 * (2.0 * (F * (B * B))))) / t_0;
	} else if (B <= 4.5e-242) {
		tmp = -Math.sqrt((-16.0 * ((A * A) * (C * F)))) / (Math.pow(B, 2.0) - (C * (A * 4.0)));
	} else if (B <= 4.7e-83) {
		tmp = -Math.sqrt((t_1 * (2.0 * (((-4.0 * A) * C) * F)))) / t_0;
	} else {
		tmp = (-Math.sqrt(2.0) / B) * Math.sqrt((F * (A - B)));
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (B * B) + (-4.0 * (A * C))
	t_1 = A - math.hypot(B, A)
	tmp = 0
	if B <= -1.55e+28:
		tmp = -math.sqrt((t_1 * (2.0 * (F * (B * B))))) / t_0
	elif B <= 4.5e-242:
		tmp = -math.sqrt((-16.0 * ((A * A) * (C * F)))) / (math.pow(B, 2.0) - (C * (A * 4.0)))
	elif B <= 4.7e-83:
		tmp = -math.sqrt((t_1 * (2.0 * (((-4.0 * A) * C) * F)))) / t_0
	else:
		tmp = (-math.sqrt(2.0) / B) * math.sqrt((F * (A - B)))
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) + Float64(-4.0 * Float64(A * C)))
	t_1 = Float64(A - hypot(B, A))
	tmp = 0.0
	if (B <= -1.55e+28)
		tmp = Float64(Float64(-sqrt(Float64(t_1 * Float64(2.0 * Float64(F * Float64(B * B)))))) / t_0);
	elseif (B <= 4.5e-242)
		tmp = Float64(Float64(-sqrt(Float64(-16.0 * Float64(Float64(A * A) * Float64(C * F))))) / Float64((B ^ 2.0) - Float64(C * Float64(A * 4.0))));
	elseif (B <= 4.7e-83)
		tmp = Float64(Float64(-sqrt(Float64(t_1 * Float64(2.0 * Float64(Float64(Float64(-4.0 * A) * C) * F))))) / t_0);
	else
		tmp = Float64(Float64(Float64(-sqrt(2.0)) / B) * sqrt(Float64(F * Float64(A - B))));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) + (-4.0 * (A * C));
	t_1 = A - hypot(B, A);
	tmp = 0.0;
	if (B <= -1.55e+28)
		tmp = -sqrt((t_1 * (2.0 * (F * (B * B))))) / t_0;
	elseif (B <= 4.5e-242)
		tmp = -sqrt((-16.0 * ((A * A) * (C * F)))) / ((B ^ 2.0) - (C * (A * 4.0)));
	elseif (B <= 4.7e-83)
		tmp = -sqrt((t_1 * (2.0 * (((-4.0 * A) * C) * F)))) / t_0;
	else
		tmp = (-sqrt(2.0) / B) * sqrt((F * (A - B)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(A - N[Sqrt[B ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -1.55e+28], N[((-N[Sqrt[N[(t$95$1 * N[(2.0 * N[(F * N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[B, 4.5e-242], N[((-N[Sqrt[N[(-16.0 * N[(N[(A * A), $MachinePrecision] * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[Power[B, 2.0], $MachinePrecision] - N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 4.7e-83], N[((-N[Sqrt[N[(t$95$1 * N[(2.0 * N[(N[(N[(-4.0 * A), $MachinePrecision] * C), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(N[((-N[Sqrt[2.0], $MachinePrecision]) / B), $MachinePrecision] * N[Sqrt[N[(F * N[(A - B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < -1.55e28

   1. Initial program 16.5%

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

   2. Taylor expanded in C around 0 11.3%

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

      1. unpow2 11.3%

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

      2. unpow2 11.3%

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

      3. hypot-def 13.0%

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

   4. Simplified 13.0%

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

      1. *-un-lft-identity 13.0%

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

      2. associate-*l* 13.0%

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

      3. pow2 13.0%

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

      4. associate-*l* 13.0%

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

      5. pow2 13.0%

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

      6. associate-*l* 13.0%

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

   6. Applied egg-rr 13.0%

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

      1. *-lft-identity 13.0%

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

      2. associate-*r* 13.0%

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

      3. unpow2 13.0%

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

      4. cancel-sign-sub-inv 13.0%

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

      5. unpow2 13.0%

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

      6. metadata-eval 13.0%

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

      7. unpow2 13.0%

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

      8. cancel-sign-sub-inv 13.0%

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

   8. Simplified 13.0%

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

   9. Taylor expanded in B around inf 13.4%

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

       1. unpow2 13.4%

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

   11. Simplified 13.4%

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

   if -1.55e28 < B < 4.4999999999999999e-242

   1. Initial program 22.8%

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

   2. Taylor expanded in C around 0 21.8%

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

      1. unpow2 21.8%

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

      2. unpow2 21.8%

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

      3. hypot-def 23.9%

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

   4. Simplified 23.9%

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

   5. Taylor expanded in A around -inf 21.8%

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

      1. unpow2 21.8%

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

   7. Simplified 21.8%

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

   if 4.4999999999999999e-242 < B < 4.7000000000000003e-83

   1. Initial program 12.5%

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

   2. Taylor expanded in C around 0 10.4%

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

      1. unpow2 10.4%

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

      2. unpow2 10.4%

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

      3. hypot-def 14.9%

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

   4. Simplified 14.9%

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

      1. *-un-lft-identity 14.9%

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

      2. associate-*l* 14.9%

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

      3. pow2 14.9%

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

      4. associate-*l* 14.9%

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

      5. pow2 14.9%

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

      6. associate-*l* 14.9%

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

   6. Applied egg-rr 14.9%

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

      1. *-lft-identity 14.9%

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

      2. associate-*r* 14.9%

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

      3. unpow2 14.9%

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

      4. cancel-sign-sub-inv 14.9%

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

      5. unpow2 14.9%

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

      6. metadata-eval 14.9%

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

      7. unpow2 14.9%

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

      8. cancel-sign-sub-inv 14.9%

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

   8. Simplified 14.9%

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

   9. Taylor expanded in B around 0 14.9%

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

       1. associate-*r* 14.9%

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

   11. Simplified 14.9%

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

   if 4.7000000000000003e-83 < B

   1. Initial program 20.2%

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

   2. Taylor expanded in B around inf 14.6%

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

   3. Taylor expanded in C around 0 44.9%

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

      1. associate-*r* 44.9%

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

      2. neg-mul-1 44.9%

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

      3. distribute-neg-frac 44.9%

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

   5. Simplified 44.9%

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

4. Final simplification 25.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -1.55 \cdot 10^{+28}:\\ \;\;\;\;\frac{-\sqrt{\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot \left(2 \cdot \left(F \cdot \left(B \cdot B\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 4.5 \cdot 10^{-242}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(\left(A \cdot A\right) \cdot \left(C \cdot F\right)\right)}}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;B \leq 4.7 \cdot 10^{-83}:\\ \;\;\;\;\frac{-\sqrt{\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot \left(2 \cdot \left(\left(\left(-4 \cdot A\right) \cdot C\right) \cdot F\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - B\right)}\\ \end{array} \]

Alternative 8: 27.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} \mathbf{if}\;B \leq -1.6 \cdot 10^{+28}:\\ \;\;\;\;\frac{-\sqrt{\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot \left(2 \cdot \left(F \cdot \left(B \cdot B\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 125000:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(\left(A \cdot A\right) \cdot \left(C \cdot F\right)\right)}}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - B\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (if (<= B -1.6e+28)
   (/
    (- (sqrt (* (- A (hypot B A)) (* 2.0 (* F (* B B))))))
    (+ (* B B) (* -4.0 (* A C))))
   (if (<= B 125000.0)
     (/
      (- (sqrt (* -16.0 (* (* A A) (* C F)))))
      (- (pow B 2.0) (* C (* A 4.0))))
     (* (/ (- (sqrt 2.0)) B) (sqrt (* F (- A B)))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double tmp;
	if (B <= -1.6e+28) {
		tmp = -sqrt(((A - hypot(B, A)) * (2.0 * (F * (B * B))))) / ((B * B) + (-4.0 * (A * C)));
	} else if (B <= 125000.0) {
		tmp = -sqrt((-16.0 * ((A * A) * (C * F)))) / (pow(B, 2.0) - (C * (A * 4.0)));
	} else {
		tmp = (-sqrt(2.0) / B) * sqrt((F * (A - B)));
	}
	return tmp;
}

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double tmp;
	if (B <= -1.6e+28) {
		tmp = -Math.sqrt(((A - Math.hypot(B, A)) * (2.0 * (F * (B * B))))) / ((B * B) + (-4.0 * (A * C)));
	} else if (B <= 125000.0) {
		tmp = -Math.sqrt((-16.0 * ((A * A) * (C * F)))) / (Math.pow(B, 2.0) - (C * (A * 4.0)));
	} else {
		tmp = (-Math.sqrt(2.0) / B) * Math.sqrt((F * (A - B)));
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	tmp = 0
	if B <= -1.6e+28:
		tmp = -math.sqrt(((A - math.hypot(B, A)) * (2.0 * (F * (B * B))))) / ((B * B) + (-4.0 * (A * C)))
	elif B <= 125000.0:
		tmp = -math.sqrt((-16.0 * ((A * A) * (C * F)))) / (math.pow(B, 2.0) - (C * (A * 4.0)))
	else:
		tmp = (-math.sqrt(2.0) / B) * math.sqrt((F * (A - B)))
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	tmp = 0.0
	if (B <= -1.6e+28)
		tmp = Float64(Float64(-sqrt(Float64(Float64(A - hypot(B, A)) * Float64(2.0 * Float64(F * Float64(B * B)))))) / Float64(Float64(B * B) + Float64(-4.0 * Float64(A * C))));
	elseif (B <= 125000.0)
		tmp = Float64(Float64(-sqrt(Float64(-16.0 * Float64(Float64(A * A) * Float64(C * F))))) / Float64((B ^ 2.0) - Float64(C * Float64(A * 4.0))));
	else
		tmp = Float64(Float64(Float64(-sqrt(2.0)) / B) * sqrt(Float64(F * Float64(A - B))));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (B <= -1.6e+28)
		tmp = -sqrt(((A - hypot(B, A)) * (2.0 * (F * (B * B))))) / ((B * B) + (-4.0 * (A * C)));
	elseif (B <= 125000.0)
		tmp = -sqrt((-16.0 * ((A * A) * (C * F)))) / ((B ^ 2.0) - (C * (A * 4.0)));
	else
		tmp = (-sqrt(2.0) / B) * sqrt((F * (A - B)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := If[LessEqual[B, -1.6e+28], N[((-N[Sqrt[N[(N[(A - N[Sqrt[B ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(F * N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[(B * B), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 125000.0], N[((-N[Sqrt[N[(-16.0 * N[(N[(A * A), $MachinePrecision] * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[Power[B, 2.0], $MachinePrecision] - N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-N[Sqrt[2.0], $MachinePrecision]) / B), $MachinePrecision] * N[Sqrt[N[(F * N[(A - B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < -1.6e28

   1. Initial program 16.5%

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

   2. Taylor expanded in C around 0 11.3%

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

      1. unpow2 11.3%

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

      2. unpow2 11.3%

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

      3. hypot-def 13.0%

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

   4. Simplified 13.0%

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

      1. *-un-lft-identity 13.0%

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

      2. associate-*l* 13.0%

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

      3. pow2 13.0%

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

      4. associate-*l* 13.0%

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

      5. pow2 13.0%

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

      6. associate-*l* 13.0%

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

   6. Applied egg-rr 13.0%

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

      1. *-lft-identity 13.0%

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

      2. associate-*r* 13.0%

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

      3. unpow2 13.0%

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

      4. cancel-sign-sub-inv 13.0%

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

      5. unpow2 13.0%

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

      6. metadata-eval 13.0%

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

      7. unpow2 13.0%

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

      8. cancel-sign-sub-inv 13.0%

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

   8. Simplified 13.0%

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

   9. Taylor expanded in B around inf 13.4%

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

       1. unpow2 13.4%

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

   11. Simplified 13.4%

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

   if -1.6e28 < B < 125000

   1. Initial program 21.7%

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

   2. Taylor expanded in C around 0 20.5%

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

      1. unpow2 20.5%

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

      2. unpow2 20.5%

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

      3. hypot-def 23.1%

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

   4. Simplified 23.1%

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

   5. Taylor expanded in A around -inf 16.1%

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

      1. unpow2 16.1%

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

   7. Simplified 16.1%

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

   if 125000 < B

   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. Taylor expanded in B around inf 10.8%

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

   3. Taylor expanded in C around 0 54.2%

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

      1. associate-*r* 54.2%

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

      2. neg-mul-1 54.2%

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

      3. distribute-neg-frac 54.2%

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

   5. Simplified 54.2%

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

4. Final simplification 23.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -1.6 \cdot 10^{+28}:\\ \;\;\;\;\frac{-\sqrt{\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot \left(2 \cdot \left(F \cdot \left(B \cdot B\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 125000:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(\left(A \cdot A\right) \cdot \left(C \cdot F\right)\right)}}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - B\right)}\\ \end{array} \]

Alternative 9: 25.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} \mathbf{if}\;B \leq 52000:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(\left(A \cdot A\right) \cdot \left(C \cdot F\right)\right)}}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - B\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (if (<= B 52000.0)
   (/ (- (sqrt (* -16.0 (* (* A A) (* C F))))) (- (pow B 2.0) (* C (* A 4.0))))
   (* (/ (- (sqrt 2.0)) B) (sqrt (* F (- A B))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double tmp;
	if (B <= 52000.0) {
		tmp = -sqrt((-16.0 * ((A * A) * (C * F)))) / (pow(B, 2.0) - (C * (A * 4.0)));
	} else {
		tmp = (-sqrt(2.0) / B) * sqrt((F * (A - B)));
	}
	return tmp;
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b <= 52000.0d0) then
        tmp = -sqrt(((-16.0d0) * ((a * a) * (c * f)))) / ((b ** 2.0d0) - (c * (a * 4.0d0)))
    else
        tmp = (-sqrt(2.0d0) / b) * sqrt((f * (a - b)))
    end if
    code = tmp
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double tmp;
	if (B <= 52000.0) {
		tmp = -Math.sqrt((-16.0 * ((A * A) * (C * F)))) / (Math.pow(B, 2.0) - (C * (A * 4.0)));
	} else {
		tmp = (-Math.sqrt(2.0) / B) * Math.sqrt((F * (A - B)));
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	tmp = 0
	if B <= 52000.0:
		tmp = -math.sqrt((-16.0 * ((A * A) * (C * F)))) / (math.pow(B, 2.0) - (C * (A * 4.0)))
	else:
		tmp = (-math.sqrt(2.0) / B) * math.sqrt((F * (A - B)))
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	tmp = 0.0
	if (B <= 52000.0)
		tmp = Float64(Float64(-sqrt(Float64(-16.0 * Float64(Float64(A * A) * Float64(C * F))))) / Float64((B ^ 2.0) - Float64(C * Float64(A * 4.0))));
	else
		tmp = Float64(Float64(Float64(-sqrt(2.0)) / B) * sqrt(Float64(F * Float64(A - B))));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (B <= 52000.0)
		tmp = -sqrt((-16.0 * ((A * A) * (C * F)))) / ((B ^ 2.0) - (C * (A * 4.0)));
	else
		tmp = (-sqrt(2.0) / B) * sqrt((F * (A - B)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := If[LessEqual[B, 52000.0], N[((-N[Sqrt[N[(-16.0 * N[(N[(A * A), $MachinePrecision] * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[Power[B, 2.0], $MachinePrecision] - N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 < 52000

   1. Initial program 20.1%

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

   2. Taylor expanded in C around 0 17.6%

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

      1. unpow2 17.6%

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

      2. unpow2 17.6%

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

      3. hypot-def 20.0%

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

   4. Simplified 20.0%

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

   5. Taylor expanded in A around -inf 12.2%

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

      1. unpow2 12.2%

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

   7. Simplified 12.2%

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

   if 52000 < B

   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. Taylor expanded in B around inf 10.8%

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

   3. Taylor expanded in C around 0 54.2%

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

      1. associate-*r* 54.2%

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

      2. neg-mul-1 54.2%

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

      3. distribute-neg-frac 54.2%

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

   5. Simplified 54.2%

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

4. Final simplification 20.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 52000:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(\left(A \cdot A\right) \cdot \left(C \cdot F\right)\right)}}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - B\right)}\\ \end{array} \]

Alternative 10: 18.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := \frac{-\sqrt{2}}{B}\\ \mathbf{if}\;B \leq 5.9 \cdot 10^{-263}:\\ \;\;\;\;t_0 \cdot \sqrt{F \cdot \frac{\left(B \cdot B\right) \cdot -0.5}{C}}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \sqrt{F \cdot \left(A - B\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt 2.0)) B)))
   (if (<= B 5.9e-263)
     (* t_0 (sqrt (* F (/ (* (* B B) -0.5) C))))
     (* t_0 (sqrt (* F (- A B)))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = -sqrt(2.0) / B;
	double tmp;
	if (B <= 5.9e-263) {
		tmp = t_0 * sqrt((F * (((B * B) * -0.5) / C)));
	} else {
		tmp = t_0 * sqrt((F * (A - B)));
	}
	return tmp;
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -sqrt(2.0d0) / b
    if (b <= 5.9d-263) then
        tmp = t_0 * sqrt((f * (((b * b) * (-0.5d0)) / c)))
    else
        tmp = t_0 * sqrt((f * (a - b)))
    end if
    code = tmp
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = -Math.sqrt(2.0) / B;
	double tmp;
	if (B <= 5.9e-263) {
		tmp = t_0 * Math.sqrt((F * (((B * B) * -0.5) / C)));
	} else {
		tmp = t_0 * Math.sqrt((F * (A - B)));
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = -math.sqrt(2.0) / B
	tmp = 0
	if B <= 5.9e-263:
		tmp = t_0 * math.sqrt((F * (((B * B) * -0.5) / C)))
	else:
		tmp = t_0 * math.sqrt((F * (A - B)))
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(-sqrt(2.0)) / B)
	tmp = 0.0
	if (B <= 5.9e-263)
		tmp = Float64(t_0 * sqrt(Float64(F * Float64(Float64(Float64(B * B) * -0.5) / C))));
	else
		tmp = Float64(t_0 * sqrt(Float64(F * Float64(A - B))));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = -sqrt(2.0) / B;
	tmp = 0.0;
	if (B <= 5.9e-263)
		tmp = t_0 * sqrt((F * (((B * B) * -0.5) / C)));
	else
		tmp = t_0 * sqrt((F * (A - B)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[((-N[Sqrt[2.0], $MachinePrecision]) / B), $MachinePrecision]}, If[LessEqual[B, 5.9e-263], N[(t$95$0 * N[Sqrt[N[(F * N[(N[(N[(B * B), $MachinePrecision] * -0.5), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Sqrt[N[(F * N[(A - B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 5.89999999999999972e-263

   1. Initial program 20.7%

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

      1. add-cbrt-cube 14.8%

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

      2. pow3 14.9%

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

      3. unpow2 14.9%

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

      4. unpow2 14.9%

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

      5. hypot-def 14.9%

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

   3. Applied egg-rr 14.9%

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

   4. Taylor expanded in A around 0 1.5%

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

      1. associate-*r* 1.5%

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

      2. neg-mul-1 1.5%

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

      3. distribute-neg-frac 1.5%

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

      4. unpow2 1.5%

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

      5. unpow2 1.5%

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

      6. hypot-def 2.4%

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

   6. Simplified 2.4%

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

   7. Taylor expanded in C around inf 6.6%

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

      1. associate-*r/ 6.6%

         \[\leadsto \frac{-\sqrt{2}}{B} \cdot \sqrt{\color{blue}{\frac{-0.5 \cdot {B}^{2}}{C}} \cdot F} \]

      2. unpow2 6.6%

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

   9. Simplified 6.6%

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

   if 5.89999999999999972e-263 < B

   1. Initial program 16.9%

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

   2. Taylor expanded in B around inf 11.0%

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

   3. Taylor expanded in C around 0 30.6%

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

      1. associate-*r* 30.6%

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

      2. neg-mul-1 30.6%

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

      3. distribute-neg-frac 30.6%

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

   5. Simplified 30.6%

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

4. Final simplification 17.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 5.9 \cdot 10^{-263}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \frac{\left(B \cdot B\right) \cdot -0.5}{C}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - B\right)}\\ \end{array} \]

Alternative 11: 15.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq 9.5 \cdot 10^{-226}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(A + \left(C - B\right)\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - B\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (+ (* B B) (* -4.0 (* A C)))))
   (if (<= B 9.5e-226)
     (/ (- (sqrt (* 2.0 (* t_0 (* F (+ A (- C B))))))) t_0)
     (* (/ (- (sqrt 2.0)) B) (sqrt (* F (- A B)))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) + (-4.0 * (A * C));
	double tmp;
	if (B <= 9.5e-226) {
		tmp = -sqrt((2.0 * (t_0 * (F * (A + (C - B)))))) / t_0;
	} else {
		tmp = (-sqrt(2.0) / B) * sqrt((F * (A - B)));
	}
	return tmp;
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b * b) + ((-4.0d0) * (a * c))
    if (b <= 9.5d-226) then
        tmp = -sqrt((2.0d0 * (t_0 * (f * (a + (c - b)))))) / t_0
    else
        tmp = (-sqrt(2.0d0) / b) * sqrt((f * (a - b)))
    end if
    code = tmp
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) + (-4.0 * (A * C));
	double tmp;
	if (B <= 9.5e-226) {
		tmp = -Math.sqrt((2.0 * (t_0 * (F * (A + (C - B)))))) / t_0;
	} else {
		tmp = (-Math.sqrt(2.0) / B) * Math.sqrt((F * (A - B)));
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (B * B) + (-4.0 * (A * C))
	tmp = 0
	if B <= 9.5e-226:
		tmp = -math.sqrt((2.0 * (t_0 * (F * (A + (C - B)))))) / t_0
	else:
		tmp = (-math.sqrt(2.0) / B) * math.sqrt((F * (A - B)))
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) + Float64(-4.0 * Float64(A * C)))
	tmp = 0.0
	if (B <= 9.5e-226)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_0 * Float64(F * Float64(A + Float64(C - B))))))) / t_0);
	else
		tmp = Float64(Float64(Float64(-sqrt(2.0)) / B) * sqrt(Float64(F * Float64(A - B))));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) + (-4.0 * (A * C));
	tmp = 0.0;
	if (B <= 9.5e-226)
		tmp = -sqrt((2.0 * (t_0 * (F * (A + (C - B)))))) / t_0;
	else
		tmp = (-sqrt(2.0) / B) * sqrt((F * (A - B)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, 9.5e-226], N[((-N[Sqrt[N[(2.0 * N[(t$95$0 * N[(F * N[(A + N[(C - B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(N[((-N[Sqrt[2.0], $MachinePrecision]) / B), $MachinePrecision] * N[Sqrt[N[(F * N[(A - B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 9.5000000000000007e-226

   1. Initial program 19.6%

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

   2. Taylor expanded in B around inf 4.6%

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

      1. *-un-lft-identity 4.6%

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

      2. associate-*l* 4.6%

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

      3. pow2 4.6%

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

      4. associate-*l* 4.6%

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

      5. associate--l+ 4.6%

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

      6. pow2 4.6%

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

      7. associate-*l* 4.6%

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

   4. Applied egg-rr 4.6%

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

      1. *-lft-identity 4.6%

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

      2. associate-*l* 4.8%

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

      3. unpow2 4.8%

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

      4. cancel-sign-sub-inv 4.8%

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

      5. unpow2 4.8%

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

      6. metadata-eval 4.8%

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

      7. unpow2 4.8%

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

      8. cancel-sign-sub-inv 4.8%

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

      9. unpow2 4.8%

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

      10. metadata-eval 4.8%

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

   6. Simplified 4.8%

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

   if 9.5000000000000007e-226 < B

   1. Initial program 18.0%

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

   2. Taylor expanded in B around inf 11.4%

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

   3. Taylor expanded in C around 0 32.8%

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

      1. associate-*r* 32.8%

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

      2. neg-mul-1 32.8%

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

      3. distribute-neg-frac 32.8%

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

   5. Simplified 32.8%

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

4. Final simplification 16.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 9.5 \cdot 10^{-226}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(A + \left(C - B\right)\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - B\right)}\\ \end{array} \]

Alternative 12: 15.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq 4.3 \cdot 10^{-226}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(A + \left(C - B\right)\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(-B\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (+ (* B B) (* -4.0 (* A C)))))
   (if (<= B 4.3e-226)
     (/ (- (sqrt (* 2.0 (* t_0 (* F (+ A (- C B))))))) t_0)
     (* (/ (- (sqrt 2.0)) B) (sqrt (* F (- B)))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) + (-4.0 * (A * C));
	double tmp;
	if (B <= 4.3e-226) {
		tmp = -sqrt((2.0 * (t_0 * (F * (A + (C - B)))))) / t_0;
	} else {
		tmp = (-sqrt(2.0) / B) * sqrt((F * -B));
	}
	return tmp;
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b * b) + ((-4.0d0) * (a * c))
    if (b <= 4.3d-226) then
        tmp = -sqrt((2.0d0 * (t_0 * (f * (a + (c - b)))))) / t_0
    else
        tmp = (-sqrt(2.0d0) / b) * sqrt((f * -b))
    end if
    code = tmp
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) + (-4.0 * (A * C));
	double tmp;
	if (B <= 4.3e-226) {
		tmp = -Math.sqrt((2.0 * (t_0 * (F * (A + (C - B)))))) / t_0;
	} else {
		tmp = (-Math.sqrt(2.0) / B) * Math.sqrt((F * -B));
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (B * B) + (-4.0 * (A * C))
	tmp = 0
	if B <= 4.3e-226:
		tmp = -math.sqrt((2.0 * (t_0 * (F * (A + (C - B)))))) / t_0
	else:
		tmp = (-math.sqrt(2.0) / B) * math.sqrt((F * -B))
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) + Float64(-4.0 * Float64(A * C)))
	tmp = 0.0
	if (B <= 4.3e-226)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_0 * Float64(F * Float64(A + Float64(C - B))))))) / t_0);
	else
		tmp = Float64(Float64(Float64(-sqrt(2.0)) / B) * sqrt(Float64(F * Float64(-B))));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) + (-4.0 * (A * C));
	tmp = 0.0;
	if (B <= 4.3e-226)
		tmp = -sqrt((2.0 * (t_0 * (F * (A + (C - B)))))) / t_0;
	else
		tmp = (-sqrt(2.0) / B) * sqrt((F * -B));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, 4.3e-226], N[((-N[Sqrt[N[(2.0 * N[(t$95$0 * N[(F * N[(A + N[(C - B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(N[((-N[Sqrt[2.0], $MachinePrecision]) / B), $MachinePrecision] * N[Sqrt[N[(F * (-B)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 4.30000000000000024e-226

   1. Initial program 19.6%

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

   2. Taylor expanded in B around inf 4.6%

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

      1. *-un-lft-identity 4.6%

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

      2. associate-*l* 4.6%

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

      3. pow2 4.6%

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

      4. associate-*l* 4.6%

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

      5. associate--l+ 4.6%

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

      6. pow2 4.6%

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

      7. associate-*l* 4.6%

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

   4. Applied egg-rr 4.6%

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

      1. *-lft-identity 4.6%

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

      2. associate-*l* 4.8%

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

      3. unpow2 4.8%

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

      4. cancel-sign-sub-inv 4.8%

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

      5. unpow2 4.8%

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

      6. metadata-eval 4.8%

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

      7. unpow2 4.8%

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

      8. cancel-sign-sub-inv 4.8%

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

      9. unpow2 4.8%

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

      10. metadata-eval 4.8%

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

   6. Simplified 4.8%

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

   if 4.30000000000000024e-226 < B

   1. Initial program 18.0%

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

      1. add-cbrt-cube 13.6%

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

      2. pow3 13.5%

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

      3. unpow2 13.5%

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

      4. unpow2 13.5%

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

      5. hypot-def 13.5%

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

   3. Applied egg-rr 13.5%

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

   4. Taylor expanded in A around 0 18.4%

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

      1. associate-*r* 18.4%

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

      2. neg-mul-1 18.4%

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

      3. distribute-neg-frac 18.4%

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

      4. unpow2 18.4%

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

      5. unpow2 18.4%

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

      6. hypot-def 36.8%

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

   6. Simplified 36.8%

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

   7. Taylor expanded in C around 0 33.2%

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

      1. mul-1-neg 33.2%

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

   9. Simplified 33.2%

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

4. Final simplification 16.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 4.3 \cdot 10^{-226}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(A + \left(C - B\right)\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(-B\right)}\\ \end{array} \]

Alternative 13: 7.7% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: A and C should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    t_0 = (b * b) - ((a * c) * 4.0d0)
    code = -sqrt((2.0d0 * ((a + (c - b)) * (f * t_0)))) / t_0
end function

Java

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

Python

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

Julia

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

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp = code(A, B, C, F)
	t_0 = (B * B) - ((A * C) * 4.0);
	tmp = -sqrt((2.0 * ((A + (C - B)) * (F * t_0)))) / t_0;
end

Wolfram

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

Derivation

1. Initial program 18.9%

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

2. Taylor expanded in B around inf 7.5%

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

   1. distribute-frac-neg 7.5%

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

   2. associate-*l* 7.5%

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

   3. pow2 7.5%

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

   4. associate-*l* 7.5%

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

   5. associate--l+ 7.5%

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

   6. pow2 7.5%

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

   7. associate-*l* 7.5%

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

4. Applied egg-rr 7.5%

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

5. Final simplification 7.5%

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

Alternative 14: 7.7% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: A and C should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    t_0 = (b * b) + ((-4.0d0) * (a * c))
    code = -sqrt((2.0d0 * (t_0 * (f * (a + (c - b)))))) / t_0
end function

Java

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

Python

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

Julia

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

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp = code(A, B, C, F)
	t_0 = (B * B) + (-4.0 * (A * C));
	tmp = -sqrt((2.0 * (t_0 * (F * (A + (C - B)))))) / t_0;
end

Wolfram

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

Derivation

1. Initial program 18.9%

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

2. Taylor expanded in B around inf 7.5%

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

   1. *-un-lft-identity 7.5%

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

   2. associate-*l* 7.5%

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

   3. pow2 7.5%

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

   4. associate-*l* 7.5%

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

   5. associate--l+ 7.5%

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

   6. pow2 7.5%

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

   7. associate-*l* 7.5%

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

4. Applied egg-rr 7.5%

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

   1. *-lft-identity 7.5%

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

   2. associate-*l* 7.7%

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

   3. unpow2 7.7%

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

   4. cancel-sign-sub-inv 7.7%

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

   5. unpow2 7.7%

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

   6. metadata-eval 7.7%

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

   7. unpow2 7.7%

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

   8. cancel-sign-sub-inv 7.7%

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

   9. unpow2 7.7%

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

   10. metadata-eval 7.7%

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

6. Simplified 7.7%

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

7. Final simplification 7.7%

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

Alternative 15: 0.8% accurate, 5.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp = code(A, B, C, F)
	tmp = (sqrt((C * F)) * (B * -2.0)) / ((B * B) + (-4.0 * (A * C)));
end

Wolfram

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

Derivation

1. Initial program 18.9%

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

2. Taylor expanded in C around 0 16.1%

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

   1. unpow2 16.1%

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

   2. unpow2 16.1%

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

   3. hypot-def 18.3%

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

4. Simplified 18.3%

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

   1. *-un-lft-identity 18.3%

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

   2. associate-*l* 18.2%

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

   3. pow2 18.2%

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

   4. associate-*l* 18.2%

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

   5. pow2 18.2%

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

   6. associate-*l* 18.2%

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

6. Applied egg-rr 18.2%

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

   1. *-lft-identity 18.2%

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

   2. associate-*r* 18.3%

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

   3. unpow2 18.3%

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

   4. cancel-sign-sub-inv 18.3%

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

   5. unpow2 18.3%

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

   6. metadata-eval 18.3%

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

   7. unpow2 18.3%

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

   8. cancel-sign-sub-inv 18.3%

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

8. Simplified 18.3%

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

9. Taylor expanded in B around 0 2.8%

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

    1. unpow2 2.8%

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

    2. rem-square-sqrt 2.8%

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

11. Simplified 2.8%

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

12. Final simplification 2.8%

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

Reproduce

herbie shell --seed 2023187 
(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))))