ABCF->ab-angle a

Percentage Accurate: 19.7% → 52.2%

Time: 29.7s

Alternatives: 15

Speedup: 3.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 19.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 52.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;{B_m}^{2} \leq 2 \cdot 10^{+191}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B_m, B_m, \left(C \cdot A\right) \cdot -4\right)\right)} \cdot \left(-\sqrt{\left(C + A\right) + \mathsf{hypot}\left(A - C, B_m\right)}\right)}{{B_m}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(B_m, C\right)} \cdot \sqrt{F}\right) \cdot \left(\sqrt{2} \cdot \frac{-1}{B_m}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 2e+191)
   (/
    (*
     (sqrt (* 2.0 (* F (fma B_m B_m (* (* C A) -4.0)))))
     (- (sqrt (+ (+ C A) (hypot (- A C) B_m)))))
    (- (pow B_m 2.0) (* C (* A 4.0))))
   (* (* (sqrt (+ C (hypot B_m C))) (sqrt F)) (* (sqrt 2.0) (/ -1.0 B_m)))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 2e+191) {
		tmp = (sqrt((2.0 * (F * fma(B_m, B_m, ((C * A) * -4.0))))) * -sqrt(((C + A) + hypot((A - C), B_m)))) / (pow(B_m, 2.0) - (C * (A * 4.0)));
	} else {
		tmp = (sqrt((C + hypot(B_m, C))) * sqrt(F)) * (sqrt(2.0) * (-1.0 / B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e+191)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(F * fma(B_m, B_m, Float64(Float64(C * A) * -4.0))))) * Float64(-sqrt(Float64(Float64(C + A) + hypot(Float64(A - C), B_m))))) / Float64((B_m ^ 2.0) - Float64(C * Float64(A * 4.0))));
	else
		tmp = Float64(Float64(sqrt(Float64(C + hypot(B_m, C))) * sqrt(F)) * Float64(sqrt(2.0) * Float64(-1.0 / B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+191], N[(N[(N[Sqrt[N[(2.0 * N[(F * N[(B$95$m * B$95$m + N[(N[(C * A), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(N[(C + A), $MachinePrecision] + N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-1.0 / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B 2) < 2.00000000000000015e191

   1. Initial program 26.3%

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

      1. sqrt-prod 30.1%

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

      2. associate-*r* 30.1%

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

      3. associate-*l* 30.1%

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

      4. associate-+l+ 30.4%

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

      5. unpow2 30.4%

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

      6. unpow2 30.4%

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

      7. hypot-def 42.7%

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

   3. Applied egg-rr 42.7%

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

      1. associate-*l* 42.7%

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

      2. *-commutative 42.7%

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

      3. unpow2 42.7%

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

      4. fma-neg 42.7%

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

      5. distribute-lft-neg-in 42.7%

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

      6. metadata-eval 42.7%

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

      7. *-commutative 42.7%

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

      8. *-commutative 42.7%

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

      9. associate-+r+ 42.1%

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

      10. +-commutative 42.1%

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

   5. Simplified 42.1%

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

   if 2.00000000000000015e191 < (pow.f64 B 2)

   1. Initial program 4.7%

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

   2. Taylor expanded in A around 0 2.0%

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

      1. mul-1-neg 2.0%

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

      2. *-commutative 2.0%

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

      3. distribute-rgt-neg-in 2.0%

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

      4. unpow2 2.0%

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

      5. unpow2 2.0%

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

      6. hypot-def 18.4%

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

   4. Simplified 18.4%

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

      1. pow1/2 18.5%

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

      2. *-commutative 18.5%

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

      3. unpow-prod-down 28.8%

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

      4. pow1/2 28.8%

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

      5. pow1/2 28.8%

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

   6. Applied egg-rr 28.8%

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

      1. div-inv 28.9%

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

   8. Applied egg-rr 28.9%

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

4. Final simplification 37.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{+191}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \left(C \cdot A\right) \cdot -4\right)\right)} \cdot \left(-\sqrt{\left(C + A\right) + \mathsf{hypot}\left(A - C, B\right)}\right)}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \sqrt{F}\right) \cdot \left(\sqrt{2} \cdot \frac{-1}{B}\right)\\ \end{array} \]

Alternative 2: 40.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \frac{\sqrt{2}}{B_m}\\ t_2 := -t_1\\ \mathbf{if}\;{B_m}^{2} \leq 5 \cdot 10^{-318}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;{B_m}^{2} \leq 1.1 \cdot 10^{-85}:\\ \;\;\;\;\frac{-\sqrt{\left(t_0 \cdot \left(2 \cdot F\right)\right) \cdot \left(A + A\right)}}{t_0}\\ \mathbf{elif}\;{B_m}^{2} \leq 4 \cdot 10^{-36}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \left(-\sqrt{F \cdot \left(A + \mathsf{hypot}\left(B_m, A\right)\right)}\right)}{B_m}\\ \mathbf{elif}\;{B_m}^{2} \leq 0.001:\\ \;\;\;\;t_2 \cdot \sqrt{F \cdot \left(\frac{{B_m}^{2}}{C} \cdot -0.5\right)}\\ \mathbf{elif}\;{B_m}^{2} \leq 2 \cdot 10^{+97}:\\ \;\;\;\;t_1 \cdot \left(-\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B_m, C\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \left(\sqrt{F} \cdot \sqrt{B_m}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0))))
        (t_1 (/ (sqrt 2.0) B_m))
        (t_2 (- t_1)))
   (if (<= (pow B_m 2.0) 5e-318)
     (sqrt (/ (- F) A))
     (if (<= (pow B_m 2.0) 1.1e-85)
       (/ (- (sqrt (* (* t_0 (* 2.0 F)) (+ A A)))) t_0)
       (if (<= (pow B_m 2.0) 4e-36)
         (/ (* (sqrt 2.0) (- (sqrt (* F (+ A (hypot B_m A)))))) B_m)
         (if (<= (pow B_m 2.0) 0.001)
           (* t_2 (sqrt (* F (* (/ (pow B_m 2.0) C) -0.5))))
           (if (<= (pow B_m 2.0) 2e+97)
             (* t_1 (- (sqrt (* F (+ C (hypot B_m C))))))
             (* t_2 (* (sqrt F) (sqrt B_m))))))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_1 = sqrt(2.0) / B_m;
	double t_2 = -t_1;
	double tmp;
	if (pow(B_m, 2.0) <= 5e-318) {
		tmp = sqrt((-F / A));
	} else if (pow(B_m, 2.0) <= 1.1e-85) {
		tmp = -sqrt(((t_0 * (2.0 * F)) * (A + A))) / t_0;
	} else if (pow(B_m, 2.0) <= 4e-36) {
		tmp = (sqrt(2.0) * -sqrt((F * (A + hypot(B_m, A))))) / B_m;
	} else if (pow(B_m, 2.0) <= 0.001) {
		tmp = t_2 * sqrt((F * ((pow(B_m, 2.0) / C) * -0.5)));
	} else if (pow(B_m, 2.0) <= 2e+97) {
		tmp = t_1 * -sqrt((F * (C + hypot(B_m, C))));
	} else {
		tmp = t_2 * (sqrt(F) * sqrt(B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_1 = Float64(sqrt(2.0) / B_m)
	t_2 = Float64(-t_1)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e-318)
		tmp = sqrt(Float64(Float64(-F) / A));
	elseif ((B_m ^ 2.0) <= 1.1e-85)
		tmp = Float64(Float64(-sqrt(Float64(Float64(t_0 * Float64(2.0 * F)) * Float64(A + A)))) / t_0);
	elseif ((B_m ^ 2.0) <= 4e-36)
		tmp = Float64(Float64(sqrt(2.0) * Float64(-sqrt(Float64(F * Float64(A + hypot(B_m, A)))))) / B_m);
	elseif ((B_m ^ 2.0) <= 0.001)
		tmp = Float64(t_2 * sqrt(Float64(F * Float64(Float64((B_m ^ 2.0) / C) * -0.5))));
	elseif ((B_m ^ 2.0) <= 2e+97)
		tmp = Float64(t_1 * Float64(-sqrt(Float64(F * Float64(C + hypot(B_m, C))))));
	else
		tmp = Float64(t_2 * Float64(sqrt(F) * sqrt(B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision]}, Block[{t$95$2 = (-t$95$1)}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-318], N[Sqrt[N[((-F) / A), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1.1e-85], N[((-N[Sqrt[N[(N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision] * N[(A + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e-36], N[(N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F * N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / B$95$m), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 0.001], N[(t$95$2 * N[Sqrt[N[(F * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+97], N[(t$95$1 * (-N[Sqrt[N[(F * N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(t$95$2 * N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (pow.f64 B 2) < 4.9999987e-318

   1. Initial program 13.1%

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

      1. add-sqr-sqrt 6.1%

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

      2. sqrt-unprod 4.6%

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

      3. frac-times 2.4%

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

   3. Applied egg-rr 3.6%

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

      1. associate-/l* 4.1%

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

      2. associate-*l* 4.1%

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

      3. *-commutative 4.1%

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

      4. unpow2 4.1%

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

      5. fma-neg 4.1%

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

      6. distribute-lft-neg-in 4.1%

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

      7. metadata-eval 4.1%

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

      8. *-commutative 4.1%

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

      9. *-commutative 4.1%

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

   5. Simplified 3.2%

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

   6. Taylor expanded in C around inf 25.7%

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

      1. mul-1-neg 25.7%

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

   8. Simplified 25.7%

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

   if 4.9999987e-318 < (pow.f64 B 2) < 1.1e-85

   1. Initial program 34.4%

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

   3. Simplified 44.5%

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

   4. Taylor expanded in A around inf 38.9%

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

      1. distribute-rgt1-in 38.9%

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

      2. metadata-eval 38.9%

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

      3. mul0-lft 38.9%

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

   6. Simplified 38.9%

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

   if 1.1e-85 < (pow.f64 B 2) < 3.9999999999999998e-36

   1. Initial program 46.4%

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

   2. Taylor expanded in C around 0 10.2%

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

      1. mul-1-neg 10.2%

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

      2. distribute-rgt-neg-in 10.2%

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

      3. +-commutative 10.2%

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

      4. unpow2 10.2%

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

      5. unpow2 10.2%

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

      6. hypot-def 10.2%

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

   4. Simplified 10.2%

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

      1. associate-*l/ 10.2%

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

   6. Applied egg-rr 10.2%

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

   if 3.9999999999999998e-36 < (pow.f64 B 2) < 1e-3

   1. Initial program 2.1%

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

   2. Taylor expanded in A around 0 2.0%

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

      1. mul-1-neg 2.0%

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

      2. *-commutative 2.0%

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

      3. distribute-rgt-neg-in 2.0%

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

      4. unpow2 2.0%

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

      5. unpow2 2.0%

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

      6. hypot-def 2.8%

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

   4. Simplified 2.8%

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

   5. Taylor expanded in C around -inf 2.2%

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

      1. *-commutative 2.2%

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

   7. Simplified 2.2%

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

   if 1e-3 < (pow.f64 B 2) < 2.0000000000000001e97

   1. Initial program 42.5%

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

   2. Taylor expanded in A around 0 22.0%

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

      1. mul-1-neg 22.0%

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

      2. *-commutative 22.0%

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

      3. distribute-rgt-neg-in 22.0%

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

      4. unpow2 22.0%

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

      5. unpow2 22.0%

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

      6. hypot-def 26.5%

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

   4. Simplified 26.5%

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

   if 2.0000000000000001e97 < (pow.f64 B 2)

   1. Initial program 8.6%

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

   2. Taylor expanded in A around 0 4.6%

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

      1. mul-1-neg 4.6%

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

      2. *-commutative 4.6%

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

      3. distribute-rgt-neg-in 4.6%

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

      4. unpow2 4.6%

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

      5. unpow2 4.6%

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

      6. hypot-def 18.3%

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

   4. Simplified 18.3%

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

      1. pow1/2 18.3%

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

      2. *-commutative 18.3%

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

      3. unpow-prod-down 27.0%

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

      4. pow1/2 27.0%

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

      5. pow1/2 27.0%

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

   6. Applied egg-rr 27.0%

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

   7. Taylor expanded in C around 0 23.8%

      \[\leadsto \left(\color{blue}{\sqrt{B}} \cdot \sqrt{F}\right) \cdot \left(-\frac{\sqrt{2}}{B}\right) \]
3. Recombined 6 regimes into one program.

4. Final simplification 25.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-318}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;{B}^{2} \leq 1.1 \cdot 10^{-85}:\\ \;\;\;\;\frac{-\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(A + A\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 4 \cdot 10^{-36}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \left(-\sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)}\right)}{B}\\ \mathbf{elif}\;{B}^{2} \leq 0.001:\\ \;\;\;\;\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(\frac{{B}^{2}}{C} \cdot -0.5\right)}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+97}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{\sqrt{2}}{B}\right) \cdot \left(\sqrt{F} \cdot \sqrt{B}\right)\\ \end{array} \]

Alternative 3: 45.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B_m, B_m, \left(C \cdot A\right) \cdot -4\right)\\ \mathbf{if}\;{B_m}^{2} \leq 2 \cdot 10^{-323}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;{B_m}^{2} \leq 2 \cdot 10^{+40}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot F\right) \cdot \left(t_0 \cdot \left(\left(C + A\right) + \mathsf{hypot}\left(A - C, B_m\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(B_m, C\right)} \cdot \sqrt{F}\right) \cdot \left(\sqrt{2} \cdot \frac{-1}{B_m}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* (* C A) -4.0))))
   (if (<= (pow B_m 2.0) 2e-323)
     (sqrt (/ (- F) A))
     (if (<= (pow B_m 2.0) 2e+40)
       (/ (- (sqrt (* (* 2.0 F) (* t_0 (+ (+ C A) (hypot (- A C) B_m)))))) t_0)
       (*
        (* (sqrt (+ C (hypot B_m C))) (sqrt F))
        (* (sqrt 2.0) (/ -1.0 B_m)))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, ((C * A) * -4.0));
	double tmp;
	if (pow(B_m, 2.0) <= 2e-323) {
		tmp = sqrt((-F / A));
	} else if (pow(B_m, 2.0) <= 2e+40) {
		tmp = -sqrt(((2.0 * F) * (t_0 * ((C + A) + hypot((A - C), B_m))))) / t_0;
	} else {
		tmp = (sqrt((C + hypot(B_m, C))) * sqrt(F)) * (sqrt(2.0) * (-1.0 / B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(Float64(C * A) * -4.0))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-323)
		tmp = sqrt(Float64(Float64(-F) / A));
	elseif ((B_m ^ 2.0) <= 2e+40)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * F) * Float64(t_0 * Float64(Float64(C + A) + hypot(Float64(A - C), B_m)))))) / t_0);
	else
		tmp = Float64(Float64(sqrt(Float64(C + hypot(B_m, C))) * sqrt(F)) * Float64(sqrt(2.0) * Float64(-1.0 / B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(N[(C * A), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-323], N[Sqrt[N[((-F) / A), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+40], N[((-N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(t$95$0 * N[(N[(C + A), $MachinePrecision] + N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-1.0 / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B 2) < 1.97626e-323

   1. Initial program 13.3%

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

      1. add-sqr-sqrt 6.2%

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

      2. sqrt-unprod 4.7%

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

      3. frac-times 2.5%

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

   3. Applied egg-rr 3.5%

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

      1. associate-/l* 4.0%

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

      2. associate-*l* 4.0%

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

      3. *-commutative 4.0%

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

      4. unpow2 4.0%

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

      5. fma-neg 4.0%

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

      6. distribute-lft-neg-in 4.0%

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

      7. metadata-eval 4.0%

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

      8. *-commutative 4.0%

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

      9. *-commutative 4.0%

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

   5. Simplified 3.2%

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

   6. Taylor expanded in C around inf 26.0%

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

      1. mul-1-neg 26.0%

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

   8. Simplified 26.0%

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

   if 1.97626e-323 < (pow.f64 B 2) < 2.00000000000000006e40

   1. Initial program 32.5%

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

      1. neg-sub0 32.5%

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

      2. div-sub 32.5%

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

      3. associate-*l* 32.5%

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

   3. Applied egg-rr 41.5%

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

   5. Simplified 36.3%

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

   if 2.00000000000000006e40 < (pow.f64 B 2)

   1. Initial program 13.0%

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

   2. Taylor expanded in A around 0 6.5%

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

      1. mul-1-neg 6.5%

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

      2. *-commutative 6.5%

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

      3. distribute-rgt-neg-in 6.5%

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

      4. unpow2 6.5%

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

      5. unpow2 6.5%

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

      6. hypot-def 19.4%

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

   4. Simplified 19.4%

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

      1. pow1/2 19.4%

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

      2. *-commutative 19.4%

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

      3. unpow-prod-down 27.0%

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

      4. pow1/2 27.0%

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

      5. pow1/2 27.0%

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

   6. Applied egg-rr 27.0%

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

      1. div-inv 27.0%

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

   8. Applied egg-rr 27.0%

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

4. Final simplification 29.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-323}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+40}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot F\right) \cdot \left(\mathsf{fma}\left(B, B, \left(C \cdot A\right) \cdot -4\right) \cdot \left(\left(C + A\right) + \mathsf{hypot}\left(A - C, B\right)\right)\right)}}{\mathsf{fma}\left(B, B, \left(C \cdot A\right) \cdot -4\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \sqrt{F}\right) \cdot \left(\sqrt{2} \cdot \frac{-1}{B}\right)\\ \end{array} \]

Alternative 4: 47.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B_m}^{2} \leq 2 \cdot 10^{-323}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;{B_m}^{2} \leq 10^{+50}:\\ \;\;\;\;\frac{-\sqrt{\left(t_0 \cdot \left(2 \cdot F\right)\right) \cdot \left(A + \left(C + \mathsf{hypot}\left(B_m, A - C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(B_m, C\right)} \cdot \sqrt{F}\right) \cdot \left(\sqrt{2} \cdot \frac{-1}{B_m}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= (pow B_m 2.0) 2e-323)
     (sqrt (/ (- F) A))
     (if (<= (pow B_m 2.0) 1e+50)
       (/ (- (sqrt (* (* t_0 (* 2.0 F)) (+ A (+ C (hypot B_m (- A C))))))) t_0)
       (*
        (* (sqrt (+ C (hypot B_m C))) (sqrt F))
        (* (sqrt 2.0) (/ -1.0 B_m)))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (pow(B_m, 2.0) <= 2e-323) {
		tmp = sqrt((-F / A));
	} else if (pow(B_m, 2.0) <= 1e+50) {
		tmp = -sqrt(((t_0 * (2.0 * F)) * (A + (C + hypot(B_m, (A - C)))))) / t_0;
	} else {
		tmp = (sqrt((C + hypot(B_m, C))) * sqrt(F)) * (sqrt(2.0) * (-1.0 / B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-323)
		tmp = sqrt(Float64(Float64(-F) / A));
	elseif ((B_m ^ 2.0) <= 1e+50)
		tmp = Float64(Float64(-sqrt(Float64(Float64(t_0 * Float64(2.0 * F)) * Float64(A + Float64(C + hypot(B_m, Float64(A - C))))))) / t_0);
	else
		tmp = Float64(Float64(sqrt(Float64(C + hypot(B_m, C))) * sqrt(F)) * Float64(sqrt(2.0) * Float64(-1.0 / B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-323], N[Sqrt[N[((-F) / A), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+50], N[((-N[Sqrt[N[(N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision] * N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-1.0 / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B 2) < 1.97626e-323

   1. Initial program 13.3%

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

      1. add-sqr-sqrt 6.2%

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

      2. sqrt-unprod 4.7%

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

      3. frac-times 2.5%

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

   3. Applied egg-rr 3.5%

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

      1. associate-/l* 4.0%

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

      2. associate-*l* 4.0%

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

      3. *-commutative 4.0%

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

      4. unpow2 4.0%

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

      5. fma-neg 4.0%

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

      6. distribute-lft-neg-in 4.0%

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

      7. metadata-eval 4.0%

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

      8. *-commutative 4.0%

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

      9. *-commutative 4.0%

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

   5. Simplified 3.2%

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

   6. Taylor expanded in C around inf 26.0%

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

      1. mul-1-neg 26.0%

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

   8. Simplified 26.0%

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

   if 1.97626e-323 < (pow.f64 B 2) < 1.0000000000000001e50

   1. Initial program 32.6%

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

   3. Simplified 42.2%

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

   if 1.0000000000000001e50 < (pow.f64 B 2)

   1. Initial program 12.0%

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

   2. Taylor expanded in A around 0 6.0%

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

      1. mul-1-neg 6.0%

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

      2. *-commutative 6.0%

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

      3. distribute-rgt-neg-in 6.0%

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

      4. unpow2 6.0%

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

      5. unpow2 6.0%

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

      6. hypot-def 19.3%

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

   4. Simplified 19.3%

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

      1. pow1/2 19.4%

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

      2. *-commutative 19.4%

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

      3. unpow-prod-down 27.3%

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

      4. pow1/2 27.3%

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

      5. pow1/2 27.3%

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

   6. Applied egg-rr 27.3%

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

      1. div-inv 27.3%

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

   8. Applied egg-rr 27.3%

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

4. Final simplification 31.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-323}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;{B}^{2} \leq 10^{+50}:\\ \;\;\;\;\frac{-\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \sqrt{F}\right) \cdot \left(\sqrt{2} \cdot \frac{-1}{B}\right)\\ \end{array} \]

Alternative 5: 43.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := {B_m}^{2} - C \cdot \left(A \cdot 4\right)\\ \mathbf{if}\;{B_m}^{2} \leq 2 \cdot 10^{-323}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;{B_m}^{2} \leq 0.2:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot t_0\right)\right) \cdot \left(A + \mathsf{hypot}\left(B_m, A\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(B_m, C\right)} \cdot \sqrt{F}\right) \cdot \left(\sqrt{2} \cdot \frac{-1}{B_m}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (pow B_m 2.0) (* C (* A 4.0)))))
   (if (<= (pow B_m 2.0) 2e-323)
     (sqrt (/ (- F) A))
     (if (<= (pow B_m 2.0) 0.2)
       (/ (- (sqrt (* (* 2.0 (* F t_0)) (+ A (hypot B_m A))))) t_0)
       (*
        (* (sqrt (+ C (hypot B_m C))) (sqrt F))
        (* (sqrt 2.0) (/ -1.0 B_m)))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = pow(B_m, 2.0) - (C * (A * 4.0));
	double tmp;
	if (pow(B_m, 2.0) <= 2e-323) {
		tmp = sqrt((-F / A));
	} else if (pow(B_m, 2.0) <= 0.2) {
		tmp = -sqrt(((2.0 * (F * t_0)) * (A + hypot(B_m, A)))) / t_0;
	} else {
		tmp = (sqrt((C + hypot(B_m, C))) * sqrt(F)) * (sqrt(2.0) * (-1.0 / B_m));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double t_0 = Math.pow(B_m, 2.0) - (C * (A * 4.0));
	double tmp;
	if (Math.pow(B_m, 2.0) <= 2e-323) {
		tmp = Math.sqrt((-F / A));
	} else if (Math.pow(B_m, 2.0) <= 0.2) {
		tmp = -Math.sqrt(((2.0 * (F * t_0)) * (A + Math.hypot(B_m, A)))) / t_0;
	} else {
		tmp = (Math.sqrt((C + Math.hypot(B_m, C))) * Math.sqrt(F)) * (Math.sqrt(2.0) * (-1.0 / B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = math.pow(B_m, 2.0) - (C * (A * 4.0))
	tmp = 0
	if math.pow(B_m, 2.0) <= 2e-323:
		tmp = math.sqrt((-F / A))
	elif math.pow(B_m, 2.0) <= 0.2:
		tmp = -math.sqrt(((2.0 * (F * t_0)) * (A + math.hypot(B_m, A)))) / t_0
	else:
		tmp = (math.sqrt((C + math.hypot(B_m, C))) * math.sqrt(F)) * (math.sqrt(2.0) * (-1.0 / B_m))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64((B_m ^ 2.0) - Float64(C * Float64(A * 4.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-323)
		tmp = sqrt(Float64(Float64(-F) / A));
	elseif ((B_m ^ 2.0) <= 0.2)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(F * t_0)) * Float64(A + hypot(B_m, A))))) / t_0);
	else
		tmp = Float64(Float64(sqrt(Float64(C + hypot(B_m, C))) * sqrt(F)) * Float64(sqrt(2.0) * Float64(-1.0 / B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = (B_m ^ 2.0) - (C * (A * 4.0));
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 2e-323)
		tmp = sqrt((-F / A));
	elseif ((B_m ^ 2.0) <= 0.2)
		tmp = -sqrt(((2.0 * (F * t_0)) * (A + hypot(B_m, A)))) / t_0;
	else
		tmp = (sqrt((C + hypot(B_m, C))) * sqrt(F)) * (sqrt(2.0) * (-1.0 / B_m));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-323], N[Sqrt[N[((-F) / A), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 0.2], N[((-N[Sqrt[N[(N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-1.0 / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B 2) < 1.97626e-323

   1. Initial program 13.3%

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

      1. add-sqr-sqrt 6.2%

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

      2. sqrt-unprod 4.7%

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

      3. frac-times 2.5%

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

   3. Applied egg-rr 3.5%

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

      1. associate-/l* 4.0%

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

      2. associate-*l* 4.0%

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

      3. *-commutative 4.0%

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

      4. unpow2 4.0%

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

      5. fma-neg 4.0%

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

      6. distribute-lft-neg-in 4.0%

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

      7. metadata-eval 4.0%

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

      8. *-commutative 4.0%

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

      9. *-commutative 4.0%

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

   5. Simplified 3.2%

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

   6. Taylor expanded in C around inf 26.0%

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

      1. mul-1-neg 26.0%

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

   8. Simplified 26.0%

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

   if 1.97626e-323 < (pow.f64 B 2) < 0.20000000000000001

   1. Initial program 32.8%

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

   2. Taylor expanded in C around 0 31.3%

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

      1. +-commutative 31.3%

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

      2. unpow2 31.3%

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

      3. unpow2 31.3%

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

      4. hypot-def 34.6%

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

   4. Simplified 34.6%

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

   if 0.20000000000000001 < (pow.f64 B 2)

   1. Initial program 13.9%

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

   2. Taylor expanded in A around 0 7.7%

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

      1. mul-1-neg 7.7%

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

      2. *-commutative 7.7%

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

      3. distribute-rgt-neg-in 7.7%

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

      4. unpow2 7.7%

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

      5. unpow2 7.7%

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

      6. hypot-def 19.9%

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

   4. Simplified 19.9%

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

      1. pow1/2 19.9%

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

      2. *-commutative 19.9%

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

      3. unpow-prod-down 27.1%

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

      4. pow1/2 27.1%

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

      5. pow1/2 27.1%

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

   6. Applied egg-rr 27.1%

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

      1. div-inv 27.1%

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

   8. Applied egg-rr 27.1%

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

4. Final simplification 28.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-323}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;{B}^{2} \leq 0.2:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right)\right) \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)}}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \sqrt{F}\right) \cdot \left(\sqrt{2} \cdot \frac{-1}{B}\right)\\ \end{array} \]

Alternative 6: 43.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;F \leq 6.3 \cdot 10^{-299}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(B_m, C\right)} \cdot \sqrt{F}\right) \cdot \left(\sqrt{2} \cdot \frac{-1}{B_m}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F 6.3e-299)
   (sqrt (/ (- F) C))
   (* (* (sqrt (+ C (hypot B_m C))) (sqrt F)) (* (sqrt 2.0) (/ -1.0 B_m)))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 6.3e-299) {
		tmp = sqrt((-F / C));
	} else {
		tmp = (sqrt((C + hypot(B_m, C))) * sqrt(F)) * (sqrt(2.0) * (-1.0 / B_m));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 6.3e-299) {
		tmp = Math.sqrt((-F / C));
	} else {
		tmp = (Math.sqrt((C + Math.hypot(B_m, C))) * Math.sqrt(F)) * (Math.sqrt(2.0) * (-1.0 / B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if F <= 6.3e-299:
		tmp = math.sqrt((-F / C))
	else:
		tmp = (math.sqrt((C + math.hypot(B_m, C))) * math.sqrt(F)) * (math.sqrt(2.0) * (-1.0 / B_m))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= 6.3e-299)
		tmp = sqrt(Float64(Float64(-F) / C));
	else
		tmp = Float64(Float64(sqrt(Float64(C + hypot(B_m, C))) * sqrt(F)) * Float64(sqrt(2.0) * Float64(-1.0 / B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= 6.3e-299)
		tmp = sqrt((-F / C));
	else
		tmp = (sqrt((C + hypot(B_m, C))) * sqrt(F)) * (sqrt(2.0) * (-1.0 / B_m));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, 6.3e-299], N[Sqrt[N[((-F) / C), $MachinePrecision]], $MachinePrecision], N[(N[(N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-1.0 / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 6.29999999999999951e-299

   1. Initial program 25.4%

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

      1. add-sqr-sqrt 25.3%

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

      2. sqrt-unprod 17.8%

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

      3. frac-times 14.2%

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

   3. Applied egg-rr 20.0%

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

      1. associate-/l* 23.0%

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

      2. associate-*l* 23.0%

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

      3. *-commutative 23.0%

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

      4. unpow2 23.0%

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

      5. fma-neg 23.0%

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

      6. distribute-lft-neg-in 23.0%

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

      7. metadata-eval 23.0%

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

      8. *-commutative 23.0%

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

      9. *-commutative 23.0%

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

   5. Simplified 23.0%

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

   6. Taylor expanded in B around 0 41.0%

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

      1. mul-1-neg 41.0%

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

   8. Simplified 41.0%

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

   if 6.29999999999999951e-299 < F

   1. Initial program 17.0%

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

   2. Taylor expanded in A around 0 7.2%

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

      1. mul-1-neg 7.2%

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

      2. *-commutative 7.2%

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

      3. distribute-rgt-neg-in 7.2%

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

      4. unpow2 7.2%

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

      5. unpow2 7.2%

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

      6. hypot-def 15.7%

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

   4. Simplified 15.7%

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

      1. pow1/2 15.7%

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

      2. *-commutative 15.7%

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

      3. unpow-prod-down 20.2%

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

      4. pow1/2 20.2%

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

      5. pow1/2 20.2%

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

   6. Applied egg-rr 20.2%

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

      1. div-inv 20.2%

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

   8. Applied egg-rr 20.2%

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

4. Final simplification 23.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 6.3 \cdot 10^{-299}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \sqrt{F}\right) \cdot \left(\sqrt{2} \cdot \frac{-1}{B}\right)\\ \end{array} \]

Alternative 7: 43.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;F \leq 6.3 \cdot 10^{-299}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(B_m, C\right)} \cdot \sqrt{F}\right) \cdot \left(-\frac{\sqrt{2}}{B_m}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F 6.3e-299)
   (sqrt (/ (- F) C))
   (* (* (sqrt (+ C (hypot B_m C))) (sqrt F)) (- (/ (sqrt 2.0) B_m)))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 6.3e-299) {
		tmp = sqrt((-F / C));
	} else {
		tmp = (sqrt((C + hypot(B_m, C))) * sqrt(F)) * -(sqrt(2.0) / B_m);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 6.3e-299) {
		tmp = Math.sqrt((-F / C));
	} else {
		tmp = (Math.sqrt((C + Math.hypot(B_m, C))) * Math.sqrt(F)) * -(Math.sqrt(2.0) / B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if F <= 6.3e-299:
		tmp = math.sqrt((-F / C))
	else:
		tmp = (math.sqrt((C + math.hypot(B_m, C))) * math.sqrt(F)) * -(math.sqrt(2.0) / B_m)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= 6.3e-299)
		tmp = sqrt(Float64(Float64(-F) / C));
	else
		tmp = Float64(Float64(sqrt(Float64(C + hypot(B_m, C))) * sqrt(F)) * Float64(-Float64(sqrt(2.0) / B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= 6.3e-299)
		tmp = sqrt((-F / C));
	else
		tmp = (sqrt((C + hypot(B_m, C))) * sqrt(F)) * -(sqrt(2.0) / B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, 6.3e-299], N[Sqrt[N[((-F) / C), $MachinePrecision]], $MachinePrecision], N[(N[(N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * (-N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 6.29999999999999951e-299

   1. Initial program 25.4%

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

      1. add-sqr-sqrt 25.3%

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

      2. sqrt-unprod 17.8%

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

      3. frac-times 14.2%

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

   3. Applied egg-rr 20.0%

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

      1. associate-/l* 23.0%

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

      2. associate-*l* 23.0%

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

      3. *-commutative 23.0%

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

      4. unpow2 23.0%

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

      5. fma-neg 23.0%

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

      6. distribute-lft-neg-in 23.0%

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

      7. metadata-eval 23.0%

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

      8. *-commutative 23.0%

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

      9. *-commutative 23.0%

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

   5. Simplified 23.0%

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

   6. Taylor expanded in B around 0 41.0%

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

      1. mul-1-neg 41.0%

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

   8. Simplified 41.0%

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

   if 6.29999999999999951e-299 < F

   1. Initial program 17.0%

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

   2. Taylor expanded in A around 0 7.2%

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

      1. mul-1-neg 7.2%

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

      2. *-commutative 7.2%

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

      3. distribute-rgt-neg-in 7.2%

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

      4. unpow2 7.2%

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

      5. unpow2 7.2%

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

      6. hypot-def 15.7%

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

   4. Simplified 15.7%

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

      1. pow1/2 15.7%

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

      2. *-commutative 15.7%

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

      3. unpow-prod-down 20.2%

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

      4. pow1/2 20.2%

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

      5. pow1/2 20.2%

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

   6. Applied egg-rr 20.2%

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

4. Final simplification 23.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 6.3 \cdot 10^{-299}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \sqrt{F}\right) \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \end{array} \]

Alternative 8: 39.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \frac{\sqrt{2}}{B_m}\\ \mathbf{if}\;B_m \leq 3.55 \cdot 10^{-228}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;B_m \leq 2.6 \cdot 10^{-50}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;B_m \leq 1.15 \cdot 10^{+49}:\\ \;\;\;\;t_0 \cdot \left(-\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B_m, C\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-t_0\right) \cdot \left(\sqrt{F} \cdot \sqrt{B_m}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (/ (sqrt 2.0) B_m)))
   (if (<= B_m 3.55e-228)
     (sqrt (/ (- F) A))
     (if (<= B_m 2.6e-50)
       (sqrt (/ (- F) C))
       (if (<= B_m 1.15e+49)
         (* t_0 (- (sqrt (* F (+ C (hypot B_m C))))))
         (* (- t_0) (* (sqrt F) (sqrt B_m))))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = sqrt(2.0) / B_m;
	double tmp;
	if (B_m <= 3.55e-228) {
		tmp = sqrt((-F / A));
	} else if (B_m <= 2.6e-50) {
		tmp = sqrt((-F / C));
	} else if (B_m <= 1.15e+49) {
		tmp = t_0 * -sqrt((F * (C + hypot(B_m, C))));
	} else {
		tmp = -t_0 * (sqrt(F) * sqrt(B_m));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double t_0 = Math.sqrt(2.0) / B_m;
	double tmp;
	if (B_m <= 3.55e-228) {
		tmp = Math.sqrt((-F / A));
	} else if (B_m <= 2.6e-50) {
		tmp = Math.sqrt((-F / C));
	} else if (B_m <= 1.15e+49) {
		tmp = t_0 * -Math.sqrt((F * (C + Math.hypot(B_m, C))));
	} else {
		tmp = -t_0 * (Math.sqrt(F) * Math.sqrt(B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = math.sqrt(2.0) / B_m
	tmp = 0
	if B_m <= 3.55e-228:
		tmp = math.sqrt((-F / A))
	elif B_m <= 2.6e-50:
		tmp = math.sqrt((-F / C))
	elif B_m <= 1.15e+49:
		tmp = t_0 * -math.sqrt((F * (C + math.hypot(B_m, C))))
	else:
		tmp = -t_0 * (math.sqrt(F) * math.sqrt(B_m))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(sqrt(2.0) / B_m)
	tmp = 0.0
	if (B_m <= 3.55e-228)
		tmp = sqrt(Float64(Float64(-F) / A));
	elseif (B_m <= 2.6e-50)
		tmp = sqrt(Float64(Float64(-F) / C));
	elseif (B_m <= 1.15e+49)
		tmp = Float64(t_0 * Float64(-sqrt(Float64(F * Float64(C + hypot(B_m, C))))));
	else
		tmp = Float64(Float64(-t_0) * Float64(sqrt(F) * sqrt(B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = sqrt(2.0) / B_m;
	tmp = 0.0;
	if (B_m <= 3.55e-228)
		tmp = sqrt((-F / A));
	elseif (B_m <= 2.6e-50)
		tmp = sqrt((-F / C));
	elseif (B_m <= 1.15e+49)
		tmp = t_0 * -sqrt((F * (C + hypot(B_m, C))));
	else
		tmp = -t_0 * (sqrt(F) * sqrt(B_m));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision]}, If[LessEqual[B$95$m, 3.55e-228], N[Sqrt[N[((-F) / A), $MachinePrecision]], $MachinePrecision], If[LessEqual[B$95$m, 2.6e-50], N[Sqrt[N[((-F) / C), $MachinePrecision]], $MachinePrecision], If[LessEqual[B$95$m, 1.15e+49], N[(t$95$0 * (-N[Sqrt[N[(F * N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[((-t$95$0) * N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 3.5500000000000003e-228

   1. Initial program 18.4%

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

      1. add-sqr-sqrt 4.2%

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

      2. sqrt-unprod 3.4%

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

      3. frac-times 2.0%

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

   3. Applied egg-rr 3.9%

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

      1. associate-/l* 4.2%

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

      2. associate-*l* 4.2%

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

      3. *-commutative 4.2%

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

      4. unpow2 4.2%

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

      5. fma-neg 4.2%

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

      6. distribute-lft-neg-in 4.2%

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

      7. metadata-eval 4.2%

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

      8. *-commutative 4.2%

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

      9. *-commutative 4.2%

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

   5. Simplified 3.9%

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

   6. Taylor expanded in C around inf 11.4%

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

      1. mul-1-neg 11.4%

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

   8. Simplified 11.4%

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

   if 3.5500000000000003e-228 < B < 2.6000000000000001e-50

   1. Initial program 24.7%

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

      1. add-sqr-sqrt 17.2%

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

      2. sqrt-unprod 14.2%

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

      3. frac-times 11.7%

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

   3. Applied egg-rr 12.6%

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

      1. associate-/l* 13.0%

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

      2. associate-*l* 13.0%

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

      3. *-commutative 13.0%

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

      4. unpow2 13.0%

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

      5. fma-neg 13.0%

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

      6. distribute-lft-neg-in 13.0%

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

      7. metadata-eval 13.0%

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

      8. *-commutative 13.0%

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

      9. *-commutative 13.0%

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

   5. Simplified 12.5%

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

   6. Taylor expanded in B around 0 23.2%

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

      1. mul-1-neg 23.2%

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

   8. Simplified 23.2%

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

   if 2.6000000000000001e-50 < B < 1.15000000000000001e49

   1. Initial program 29.0%

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

   2. Taylor expanded in A around 0 34.3%

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

      1. mul-1-neg 34.3%

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

      2. *-commutative 34.3%

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

      3. distribute-rgt-neg-in 34.3%

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

      4. unpow2 34.3%

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

      5. unpow2 34.3%

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

      6. hypot-def 40.3%

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

   4. Simplified 40.3%

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

   if 1.15000000000000001e49 < B

   1. Initial program 9.5%

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

   2. Taylor expanded in A around 0 9.7%

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

      1. mul-1-neg 9.7%

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

      2. *-commutative 9.7%

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

      3. distribute-rgt-neg-in 9.7%

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

      4. unpow2 9.7%

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

      5. unpow2 9.7%

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

      6. hypot-def 43.0%

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

   4. Simplified 43.0%

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

      1. pow1/2 43.0%

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

      2. *-commutative 43.0%

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

      3. unpow-prod-down 64.4%

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

      4. pow1/2 64.4%

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

      5. pow1/2 64.4%

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

   6. Applied egg-rr 64.4%

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

   7. Taylor expanded in C around 0 60.3%

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

4. Final simplification 23.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.55 \cdot 10^{-228}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;B \leq 2.6 \cdot 10^{-50}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;B \leq 1.15 \cdot 10^{+49}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{\sqrt{2}}{B}\right) \cdot \left(\sqrt{F} \cdot \sqrt{B}\right)\\ \end{array} \]

Alternative 9: 39.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;F \leq 6.3 \cdot 10^{-299}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{\sqrt{2}}{B_m}\right) \cdot \left(\sqrt{F} \cdot \sqrt{B_m}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F 6.3e-299)
   (sqrt (/ (- F) C))
   (* (- (/ (sqrt 2.0) B_m)) (* (sqrt F) (sqrt B_m)))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 6.3e-299) {
		tmp = sqrt((-F / C));
	} else {
		tmp = -(sqrt(2.0) / B_m) * (sqrt(F) * sqrt(B_m));
	}
	return tmp;
}

Fortran

B_m = abs(B)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (f <= 6.3d-299) then
        tmp = sqrt((-f / c))
    else
        tmp = -(sqrt(2.0d0) / b_m) * (sqrt(f) * sqrt(b_m))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 6.3e-299) {
		tmp = Math.sqrt((-F / C));
	} else {
		tmp = -(Math.sqrt(2.0) / B_m) * (Math.sqrt(F) * Math.sqrt(B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if F <= 6.3e-299:
		tmp = math.sqrt((-F / C))
	else:
		tmp = -(math.sqrt(2.0) / B_m) * (math.sqrt(F) * math.sqrt(B_m))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= 6.3e-299)
		tmp = sqrt(Float64(Float64(-F) / C));
	else
		tmp = Float64(Float64(-Float64(sqrt(2.0) / B_m)) * Float64(sqrt(F) * sqrt(B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= 6.3e-299)
		tmp = sqrt((-F / C));
	else
		tmp = -(sqrt(2.0) / B_m) * (sqrt(F) * sqrt(B_m));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, 6.3e-299], N[Sqrt[N[((-F) / C), $MachinePrecision]], $MachinePrecision], N[((-N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision]) * N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 6.29999999999999951e-299

   1. Initial program 25.4%

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

      1. add-sqr-sqrt 25.3%

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

      2. sqrt-unprod 17.8%

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

      3. frac-times 14.2%

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

   3. Applied egg-rr 20.0%

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

      1. associate-/l* 23.0%

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

      2. associate-*l* 23.0%

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

      3. *-commutative 23.0%

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

      4. unpow2 23.0%

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

      5. fma-neg 23.0%

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

      6. distribute-lft-neg-in 23.0%

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

      7. metadata-eval 23.0%

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

      8. *-commutative 23.0%

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

      9. *-commutative 23.0%

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

   5. Simplified 23.0%

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

   6. Taylor expanded in B around 0 41.0%

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

      1. mul-1-neg 41.0%

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

   8. Simplified 41.0%

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

   if 6.29999999999999951e-299 < F

   1. Initial program 17.0%

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

   2. Taylor expanded in A around 0 7.2%

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

      1. mul-1-neg 7.2%

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

      2. *-commutative 7.2%

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

      3. distribute-rgt-neg-in 7.2%

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

      4. unpow2 7.2%

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

      5. unpow2 7.2%

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

      6. hypot-def 15.7%

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

   4. Simplified 15.7%

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

      1. pow1/2 15.7%

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

      2. *-commutative 15.7%

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

      3. unpow-prod-down 20.2%

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

      4. pow1/2 20.2%

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

      5. pow1/2 20.2%

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

   6. Applied egg-rr 20.2%

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

   7. Taylor expanded in C around 0 15.7%

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

4. Final simplification 19.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 6.3 \cdot 10^{-299}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{\sqrt{2}}{B}\right) \cdot \left(\sqrt{F} \cdot \sqrt{B}\right)\\ \end{array} \]

Alternative 10: 38.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;F \leq 6.3 \cdot 10^{-299}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;F \leq 2.25 \cdot 10^{+14}:\\ \;\;\;\;\left(-\frac{\sqrt{2}}{B_m}\right) \cdot \sqrt{B_m \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B_m}}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F 6.3e-299)
   (sqrt (/ (- F) C))
   (if (<= F 2.25e+14)
     (* (- (/ (sqrt 2.0) B_m)) (sqrt (* B_m F)))
     (* (sqrt 2.0) (- (sqrt (/ F B_m)))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 6.3e-299) {
		tmp = sqrt((-F / C));
	} else if (F <= 2.25e+14) {
		tmp = -(sqrt(2.0) / B_m) * sqrt((B_m * F));
	} else {
		tmp = sqrt(2.0) * -sqrt((F / B_m));
	}
	return tmp;
}

Fortran

B_m = abs(B)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (f <= 6.3d-299) then
        tmp = sqrt((-f / c))
    else if (f <= 2.25d+14) then
        tmp = -(sqrt(2.0d0) / b_m) * sqrt((b_m * f))
    else
        tmp = sqrt(2.0d0) * -sqrt((f / b_m))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 6.3e-299) {
		tmp = Math.sqrt((-F / C));
	} else if (F <= 2.25e+14) {
		tmp = -(Math.sqrt(2.0) / B_m) * Math.sqrt((B_m * F));
	} else {
		tmp = Math.sqrt(2.0) * -Math.sqrt((F / B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if F <= 6.3e-299:
		tmp = math.sqrt((-F / C))
	elif F <= 2.25e+14:
		tmp = -(math.sqrt(2.0) / B_m) * math.sqrt((B_m * F))
	else:
		tmp = math.sqrt(2.0) * -math.sqrt((F / B_m))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= 6.3e-299)
		tmp = sqrt(Float64(Float64(-F) / C));
	elseif (F <= 2.25e+14)
		tmp = Float64(Float64(-Float64(sqrt(2.0) / B_m)) * sqrt(Float64(B_m * F)));
	else
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F / B_m))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= 6.3e-299)
		tmp = sqrt((-F / C));
	elseif (F <= 2.25e+14)
		tmp = -(sqrt(2.0) / B_m) * sqrt((B_m * F));
	else
		tmp = sqrt(2.0) * -sqrt((F / B_m));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, 6.3e-299], N[Sqrt[N[((-F) / C), $MachinePrecision]], $MachinePrecision], If[LessEqual[F, 2.25e+14], N[((-N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision]) * N[Sqrt[N[(B$95$m * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < 6.29999999999999951e-299

   1. Initial program 25.4%

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

      1. add-sqr-sqrt 25.3%

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

      2. sqrt-unprod 17.8%

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

      3. frac-times 14.2%

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

   3. Applied egg-rr 20.0%

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

      1. associate-/l* 23.0%

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

      2. associate-*l* 23.0%

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

      3. *-commutative 23.0%

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

      4. unpow2 23.0%

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

      5. fma-neg 23.0%

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

      6. distribute-lft-neg-in 23.0%

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

      7. metadata-eval 23.0%

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

      8. *-commutative 23.0%

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

      9. *-commutative 23.0%

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

   5. Simplified 23.0%

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

   6. Taylor expanded in B around 0 41.0%

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

      1. mul-1-neg 41.0%

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

   8. Simplified 41.0%

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

   if 6.29999999999999951e-299 < F < 2.25e14

   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 A around 0 8.8%

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

      1. mul-1-neg 8.8%

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

      2. *-commutative 8.8%

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

      3. distribute-rgt-neg-in 8.8%

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

      4. unpow2 8.8%

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

      5. unpow2 8.8%

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

      6. hypot-def 22.8%

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

   4. Simplified 22.8%

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

   5. Taylor expanded in C around 0 18.2%

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

   if 2.25e14 < F

   1. Initial program 13.9%

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

   2. Taylor expanded in A around 0 5.6%

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

      1. mul-1-neg 5.6%

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

      2. *-commutative 5.6%

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

      3. distribute-rgt-neg-in 5.6%

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

      4. unpow2 5.6%

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

      5. unpow2 5.6%

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

      6. hypot-def 8.3%

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

   4. Simplified 8.3%

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

   5. Taylor expanded in C around 0 11.6%

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

      1. mul-1-neg 11.6%

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

   7. Simplified 11.6%

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

4. Final simplification 18.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 6.3 \cdot 10^{-299}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;F \leq 2.25 \cdot 10^{+14}:\\ \;\;\;\;\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{B \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 11: 30.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;F \leq 1.25 \cdot 10^{-212}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B_m}}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F 1.25e-212) (sqrt (/ (- F) C)) (* (sqrt 2.0) (- (sqrt (/ F B_m))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 1.25e-212) {
		tmp = sqrt((-F / C));
	} else {
		tmp = sqrt(2.0) * -sqrt((F / B_m));
	}
	return tmp;
}

Fortran

B_m = abs(B)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (f <= 1.25d-212) then
        tmp = sqrt((-f / c))
    else
        tmp = sqrt(2.0d0) * -sqrt((f / b_m))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 1.25e-212) {
		tmp = Math.sqrt((-F / C));
	} else {
		tmp = Math.sqrt(2.0) * -Math.sqrt((F / B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if F <= 1.25e-212:
		tmp = math.sqrt((-F / C))
	else:
		tmp = math.sqrt(2.0) * -math.sqrt((F / B_m))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= 1.25e-212)
		tmp = sqrt(Float64(Float64(-F) / C));
	else
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F / B_m))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= 1.25e-212)
		tmp = sqrt((-F / C));
	else
		tmp = sqrt(2.0) * -sqrt((F / B_m));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, 1.25e-212], N[Sqrt[N[((-F) / C), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 1.25000000000000011e-212

   1. Initial program 23.9%

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

      1. add-sqr-sqrt 14.6%

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

      2. sqrt-unprod 10.9%

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

      3. frac-times 8.7%

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

   3. Applied egg-rr 12.5%

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

      1. associate-/l* 14.2%

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

      2. associate-*l* 14.2%

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

      3. *-commutative 14.2%

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

      4. unpow2 14.2%

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

      5. fma-neg 14.2%

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

      6. distribute-lft-neg-in 14.2%

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

      7. metadata-eval 14.2%

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

      8. *-commutative 14.2%

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

      9. *-commutative 14.2%

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

   5. Simplified 14.2%

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

   6. Taylor expanded in B around 0 27.5%

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

      1. mul-1-neg 27.5%

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

   8. Simplified 27.5%

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

   if 1.25000000000000011e-212 < F

   1. Initial program 16.3%

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

   2. Taylor expanded in A around 0 7.3%

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

      1. mul-1-neg 7.3%

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

      2. *-commutative 7.3%

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

      3. distribute-rgt-neg-in 7.3%

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

      4. unpow2 7.3%

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

      5. unpow2 7.3%

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

      6. hypot-def 15.5%

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

   4. Simplified 15.5%

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

   5. Taylor expanded in C around 0 11.9%

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

      1. mul-1-neg 11.9%

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

   7. Simplified 11.9%

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

4. Final simplification 15.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 1.25 \cdot 10^{-212}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 12: 12.4% accurate, 5.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \sqrt{\frac{-F}{A}}\\ \mathbf{if}\;B_m \leq 1.6 \cdot 10^{-228}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B_m \leq 4.1 \cdot 10^{-32}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;B_m \leq 7.8 \cdot 10^{+15}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot C} \cdot \left(-2\right)}{B_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (sqrt (/ (- F) A))))
   (if (<= B_m 1.6e-228)
     t_0
     (if (<= B_m 4.1e-32)
       (sqrt (/ (- F) C))
       (if (<= B_m 7.8e+15) t_0 (/ (* (sqrt (* F C)) (- 2.0)) B_m))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = sqrt((-F / A));
	double tmp;
	if (B_m <= 1.6e-228) {
		tmp = t_0;
	} else if (B_m <= 4.1e-32) {
		tmp = sqrt((-F / C));
	} else if (B_m <= 7.8e+15) {
		tmp = t_0;
	} else {
		tmp = (sqrt((F * C)) * -2.0) / B_m;
	}
	return tmp;
}

Fortran

B_m = abs(B)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((-f / a))
    if (b_m <= 1.6d-228) then
        tmp = t_0
    else if (b_m <= 4.1d-32) then
        tmp = sqrt((-f / c))
    else if (b_m <= 7.8d+15) then
        tmp = t_0
    else
        tmp = (sqrt((f * c)) * -2.0d0) / b_m
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double t_0 = Math.sqrt((-F / A));
	double tmp;
	if (B_m <= 1.6e-228) {
		tmp = t_0;
	} else if (B_m <= 4.1e-32) {
		tmp = Math.sqrt((-F / C));
	} else if (B_m <= 7.8e+15) {
		tmp = t_0;
	} else {
		tmp = (Math.sqrt((F * C)) * -2.0) / B_m;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = math.sqrt((-F / A))
	tmp = 0
	if B_m <= 1.6e-228:
		tmp = t_0
	elif B_m <= 4.1e-32:
		tmp = math.sqrt((-F / C))
	elif B_m <= 7.8e+15:
		tmp = t_0
	else:
		tmp = (math.sqrt((F * C)) * -2.0) / B_m
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = sqrt(Float64(Float64(-F) / A))
	tmp = 0.0
	if (B_m <= 1.6e-228)
		tmp = t_0;
	elseif (B_m <= 4.1e-32)
		tmp = sqrt(Float64(Float64(-F) / C));
	elseif (B_m <= 7.8e+15)
		tmp = t_0;
	else
		tmp = Float64(Float64(sqrt(Float64(F * C)) * Float64(-2.0)) / B_m);
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = sqrt((-F / A));
	tmp = 0.0;
	if (B_m <= 1.6e-228)
		tmp = t_0;
	elseif (B_m <= 4.1e-32)
		tmp = sqrt((-F / C));
	elseif (B_m <= 7.8e+15)
		tmp = t_0;
	else
		tmp = (sqrt((F * C)) * -2.0) / B_m;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[Sqrt[N[((-F) / A), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[B$95$m, 1.6e-228], t$95$0, If[LessEqual[B$95$m, 4.1e-32], N[Sqrt[N[((-F) / C), $MachinePrecision]], $MachinePrecision], If[LessEqual[B$95$m, 7.8e+15], t$95$0, N[(N[(N[Sqrt[N[(F * C), $MachinePrecision]], $MachinePrecision] * (-2.0)), $MachinePrecision] / B$95$m), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.60000000000000011e-228 or 4.09999999999999975e-32 < B < 7.8e15

   1. Initial program 17.8%

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

      1. add-sqr-sqrt 4.1%

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

      2. sqrt-unprod 3.4%

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

      3. frac-times 1.9%

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

   3. Applied egg-rr 3.8%

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

      1. associate-/l* 4.6%

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

      2. associate-*l* 4.6%

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

      3. *-commutative 4.6%

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

      4. unpow2 4.6%

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

      5. fma-neg 4.6%

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

      6. distribute-lft-neg-in 4.6%

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

      7. metadata-eval 4.6%

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

      8. *-commutative 4.6%

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

      9. *-commutative 4.6%

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

   5. Simplified 4.4%

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

   6. Taylor expanded in C around inf 12.8%

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

      1. mul-1-neg 12.8%

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

   8. Simplified 12.8%

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

   if 1.60000000000000011e-228 < B < 4.09999999999999975e-32

   1. Initial program 26.3%

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

      1. add-sqr-sqrt 16.1%

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

      2. sqrt-unprod 13.3%

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

      3. frac-times 11.0%

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

   3. Applied egg-rr 11.8%

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

      1. associate-/l* 12.2%

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

      2. associate-*l* 12.2%

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

      3. *-commutative 12.2%

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

      4. unpow2 12.2%

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

      5. fma-neg 12.2%

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

      6. distribute-lft-neg-in 12.2%

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

      7. metadata-eval 12.2%

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

      8. *-commutative 12.2%

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

      9. *-commutative 12.2%

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

   5. Simplified 11.7%

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

   6. Taylor expanded in B around 0 25.0%

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

      1. mul-1-neg 25.0%

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

   8. Simplified 25.0%

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

   if 7.8e15 < B

   1. Initial program 15.2%

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

   2. Taylor expanded in A around 0 17.1%

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

      1. mul-1-neg 17.1%

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

      2. *-commutative 17.1%

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

      3. distribute-rgt-neg-in 17.1%

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

      4. unpow2 17.1%

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

      5. unpow2 17.1%

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

      6. hypot-def 46.3%

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

   4. Simplified 46.3%

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

   5. Taylor expanded in B around 0 10.3%

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

      1. mul-1-neg 10.3%

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

      2. *-commutative 10.3%

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

      3. *-commutative 10.3%

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

   7. Simplified 10.3%

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

      1. sqrt-pow2 10.4%

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

      2. metadata-eval 10.4%

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

      3. metadata-eval 10.4%

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

      4. associate-*r/ 10.4%

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

      5. *-commutative 10.4%

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

   9. Applied egg-rr 10.4%

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

4. Final simplification 13.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.6 \cdot 10^{-228}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;B \leq 4.1 \cdot 10^{-32}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;B \leq 7.8 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot C} \cdot \left(-2\right)}{B}\\ \end{array} \]

Alternative 13: 12.4% accurate, 5.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \sqrt{\frac{-F}{A}}\\ \mathbf{if}\;B_m \leq 2.9 \cdot 10^{-228}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B_m \leq 6.5 \cdot 10^{-32}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;B_m \leq 1.72 \cdot 10^{+15}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot C} \cdot \frac{-2}{B_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (sqrt (/ (- F) A))))
   (if (<= B_m 2.9e-228)
     t_0
     (if (<= B_m 6.5e-32)
       (sqrt (/ (- F) C))
       (if (<= B_m 1.72e+15) t_0 (* (sqrt (* F C)) (/ -2.0 B_m)))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = sqrt((-F / A));
	double tmp;
	if (B_m <= 2.9e-228) {
		tmp = t_0;
	} else if (B_m <= 6.5e-32) {
		tmp = sqrt((-F / C));
	} else if (B_m <= 1.72e+15) {
		tmp = t_0;
	} else {
		tmp = sqrt((F * C)) * (-2.0 / B_m);
	}
	return tmp;
}

Fortran

B_m = abs(B)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((-f / a))
    if (b_m <= 2.9d-228) then
        tmp = t_0
    else if (b_m <= 6.5d-32) then
        tmp = sqrt((-f / c))
    else if (b_m <= 1.72d+15) then
        tmp = t_0
    else
        tmp = sqrt((f * c)) * ((-2.0d0) / b_m)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double t_0 = Math.sqrt((-F / A));
	double tmp;
	if (B_m <= 2.9e-228) {
		tmp = t_0;
	} else if (B_m <= 6.5e-32) {
		tmp = Math.sqrt((-F / C));
	} else if (B_m <= 1.72e+15) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt((F * C)) * (-2.0 / B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = math.sqrt((-F / A))
	tmp = 0
	if B_m <= 2.9e-228:
		tmp = t_0
	elif B_m <= 6.5e-32:
		tmp = math.sqrt((-F / C))
	elif B_m <= 1.72e+15:
		tmp = t_0
	else:
		tmp = math.sqrt((F * C)) * (-2.0 / B_m)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = sqrt(Float64(Float64(-F) / A))
	tmp = 0.0
	if (B_m <= 2.9e-228)
		tmp = t_0;
	elseif (B_m <= 6.5e-32)
		tmp = sqrt(Float64(Float64(-F) / C));
	elseif (B_m <= 1.72e+15)
		tmp = t_0;
	else
		tmp = Float64(sqrt(Float64(F * C)) * Float64(-2.0 / B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = sqrt((-F / A));
	tmp = 0.0;
	if (B_m <= 2.9e-228)
		tmp = t_0;
	elseif (B_m <= 6.5e-32)
		tmp = sqrt((-F / C));
	elseif (B_m <= 1.72e+15)
		tmp = t_0;
	else
		tmp = sqrt((F * C)) * (-2.0 / B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[Sqrt[N[((-F) / A), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[B$95$m, 2.9e-228], t$95$0, If[LessEqual[B$95$m, 6.5e-32], N[Sqrt[N[((-F) / C), $MachinePrecision]], $MachinePrecision], If[LessEqual[B$95$m, 1.72e+15], t$95$0, N[(N[Sqrt[N[(F * C), $MachinePrecision]], $MachinePrecision] * N[(-2.0 / B$95$m), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if B < 2.9000000000000001e-228 or 6.49999999999999988e-32 < B < 1.72e15

   1. Initial program 17.8%

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

      1. add-sqr-sqrt 4.1%

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

      2. sqrt-unprod 3.4%

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

      3. frac-times 1.9%

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

   3. Applied egg-rr 3.8%

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

      1. associate-/l* 4.6%

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

      2. associate-*l* 4.6%

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

      3. *-commutative 4.6%

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

      4. unpow2 4.6%

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

      5. fma-neg 4.6%

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

      6. distribute-lft-neg-in 4.6%

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

      7. metadata-eval 4.6%

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

      8. *-commutative 4.6%

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

      9. *-commutative 4.6%

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

   5. Simplified 4.4%

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

   6. Taylor expanded in C around inf 12.8%

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

      1. mul-1-neg 12.8%

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

   8. Simplified 12.8%

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

   if 2.9000000000000001e-228 < B < 6.49999999999999988e-32

   1. Initial program 26.3%

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

      1. add-sqr-sqrt 16.1%

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

      2. sqrt-unprod 13.3%

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

      3. frac-times 11.0%

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

   3. Applied egg-rr 11.8%

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

      1. associate-/l* 12.2%

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

      2. associate-*l* 12.2%

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

      3. *-commutative 12.2%

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

      4. unpow2 12.2%

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

      5. fma-neg 12.2%

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

      6. distribute-lft-neg-in 12.2%

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

      7. metadata-eval 12.2%

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

      8. *-commutative 12.2%

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

      9. *-commutative 12.2%

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

   5. Simplified 11.7%

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

   6. Taylor expanded in B around 0 25.0%

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

      1. mul-1-neg 25.0%

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

   8. Simplified 25.0%

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

   if 1.72e15 < B

   1. Initial program 15.2%

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

   2. Taylor expanded in A around 0 17.1%

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

      1. mul-1-neg 17.1%

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

      2. *-commutative 17.1%

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

      3. distribute-rgt-neg-in 17.1%

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

      4. unpow2 17.1%

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

      5. unpow2 17.1%

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

      6. hypot-def 46.3%

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

   4. Simplified 46.3%

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

      1. pow1/2 46.3%

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

      2. *-commutative 46.3%

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

      3. unpow-prod-down 63.8%

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

      4. pow1/2 63.8%

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

      5. pow1/2 63.8%

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

   6. Applied egg-rr 63.8%

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

   7. Taylor expanded in B around 0 10.3%

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

      1. mul-1-neg 10.3%

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

      2. *-commutative 10.3%

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

      3. distribute-rgt-neg-in 10.3%

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

      4. *-commutative 10.3%

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

      5. mul-1-neg 10.3%

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

      6. unpow2 10.3%

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

      7. rem-square-sqrt 10.4%

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

      8. associate-*r/ 10.4%

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

      9. metadata-eval 10.4%

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

   9. Simplified 10.4%

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

4. Final simplification 13.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.9 \cdot 10^{-228}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;B \leq 6.5 \cdot 10^{-32}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;B \leq 1.72 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot C} \cdot \frac{-2}{B}\\ \end{array} \]

Alternative 14: 15.0% accurate, 5.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;A \leq -2.1 \cdot 10^{-15} \lor \neg \left(A \leq -2.8 \cdot 10^{-236}\right) \land A \leq 2.5 \cdot 10^{-50}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (or (<= A -2.1e-15) (and (not (<= A -2.8e-236)) (<= A 2.5e-50)))
   (sqrt (/ (- F) A))
   (sqrt (/ (- F) C))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if ((A <= -2.1e-15) || (!(A <= -2.8e-236) && (A <= 2.5e-50))) {
		tmp = sqrt((-F / A));
	} else {
		tmp = sqrt((-F / C));
	}
	return tmp;
}

Fortran

B_m = abs(B)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if ((a <= (-2.1d-15)) .or. (.not. (a <= (-2.8d-236))) .and. (a <= 2.5d-50)) then
        tmp = sqrt((-f / a))
    else
        tmp = sqrt((-f / c))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if ((A <= -2.1e-15) || (!(A <= -2.8e-236) && (A <= 2.5e-50))) {
		tmp = Math.sqrt((-F / A));
	} else {
		tmp = Math.sqrt((-F / C));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if (A <= -2.1e-15) or (not (A <= -2.8e-236) and (A <= 2.5e-50)):
		tmp = math.sqrt((-F / A))
	else:
		tmp = math.sqrt((-F / C))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if ((A <= -2.1e-15) || (!(A <= -2.8e-236) && (A <= 2.5e-50)))
		tmp = sqrt(Float64(Float64(-F) / A));
	else
		tmp = sqrt(Float64(Float64(-F) / C));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if ((A <= -2.1e-15) || (~((A <= -2.8e-236)) && (A <= 2.5e-50)))
		tmp = sqrt((-F / A));
	else
		tmp = sqrt((-F / C));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[Or[LessEqual[A, -2.1e-15], And[N[Not[LessEqual[A, -2.8e-236]], $MachinePrecision], LessEqual[A, 2.5e-50]]], N[Sqrt[N[((-F) / A), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[((-F) / C), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -2.09999999999999981e-15 or -2.79999999999999986e-236 < A < 2.49999999999999984e-50

   1. Initial program 16.0%

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

      1. add-sqr-sqrt 4.1%

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

      2. sqrt-unprod 3.7%

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

      3. frac-times 2.4%

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

   3. Applied egg-rr 3.6%

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

      1. associate-/l* 4.5%

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

      2. associate-*l* 4.5%

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

      3. *-commutative 4.5%

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

      4. unpow2 4.5%

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

      5. fma-neg 4.5%

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

      6. distribute-lft-neg-in 4.5%

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

      7. metadata-eval 4.5%

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

      8. *-commutative 4.5%

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

      9. *-commutative 4.5%

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

   5. Simplified 4.5%

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

   6. Taylor expanded in C around inf 17.1%

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

      1. mul-1-neg 17.1%

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

   8. Simplified 17.1%

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

   if -2.09999999999999981e-15 < A < -2.79999999999999986e-236 or 2.49999999999999984e-50 < A

   1. Initial program 20.5%

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

      1. add-sqr-sqrt 5.2%

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

      2. sqrt-unprod 4.1%

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

      3. frac-times 3.0%

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

   3. Applied egg-rr 4.6%

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

      1. associate-/l* 4.9%

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

      2. associate-*l* 4.9%

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

      3. *-commutative 4.9%

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

      4. unpow2 4.9%

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

      5. fma-neg 4.9%

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

      6. distribute-lft-neg-in 4.9%

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

      7. metadata-eval 4.9%

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

      8. *-commutative 4.9%

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

      9. *-commutative 4.9%

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

   5. Simplified 4.4%

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

   6. Taylor expanded in B around 0 18.3%

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

      1. mul-1-neg 18.3%

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

   8. Simplified 18.3%

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

4. Final simplification 17.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -2.1 \cdot 10^{-15} \lor \neg \left(A \leq -2.8 \cdot 10^{-236}\right) \land A \leq 2.5 \cdot 10^{-50}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \end{array} \]

Alternative 15: 11.5% accurate, 6.1× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \sqrt{\frac{-F}{A}} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F) :precision binary64 (sqrt (/ (- F) A)))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	return sqrt((-F / A));
}

Fortran

B_m = abs(B)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = sqrt((-f / a))
end function

Java

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

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	return math.sqrt((-F / A))

Julia

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

MATLAB

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

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := N[Sqrt[N[((-F) / A), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 18.3%

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

   1. add-sqr-sqrt 4.6%

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

   2. sqrt-unprod 3.9%

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

   3. frac-times 2.7%

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

3. Applied egg-rr 4.1%

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

   1. associate-/l* 4.7%

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

   2. associate-*l* 4.7%

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

   3. *-commutative 4.7%

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

   4. unpow2 4.7%

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

   5. fma-neg 4.7%

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

   6. distribute-lft-neg-in 4.7%

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

   7. metadata-eval 4.7%

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

   8. *-commutative 4.7%

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

   9. *-commutative 4.7%

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

5. Simplified 4.4%

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

6. Taylor expanded in C around inf 12.1%

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

   1. mul-1-neg 12.1%

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

8. Simplified 12.1%

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

9. Final simplification 12.1%

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

Reproduce

herbie shell --seed 2023321 
(FPCore (A B C F)
  :name "ABCF->ab-angle a"
  :precision binary64
  (/ (- (sqrt (* (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F)) (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))) (- (pow B 2.0) (* (* 4.0 A) C))))