ABCF->ab-angle a

Percentage Accurate: 18.4% → 51.9%

Time: 34.0s

Alternatives: 18

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: 18.4% 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: 51.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;{B_m}^{2} \leq 5 \cdot 10^{+99}:\\ \;\;\;\;\frac{\sqrt{A + \left(C + \mathsf{hypot}\left(A - C, B_m\right)\right)} \cdot \left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)\right)}\right)}{{B_m}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(B_m, C\right)} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{2}}{B_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 5e+99)
   (/
    (*
     (sqrt (+ A (+ C (hypot (- A C) B_m))))
     (- (sqrt (* 2.0 (* F (fma B_m B_m (* A (* C -4.0))))))))
    (- (pow B_m 2.0) (* C (* A 4.0))))
   (* (* (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 (pow(B_m, 2.0) <= 5e+99) {
		tmp = (sqrt((A + (C + hypot((A - C), B_m)))) * -sqrt((2.0 * (F * fma(B_m, B_m, (A * (C * -4.0))))))) / (pow(B_m, 2.0) - (C * (A * 4.0)));
	} else {
		tmp = (sqrt((C + hypot(B_m, C))) * sqrt(F)) * (-sqrt(2.0) / B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e+99)
		tmp = Float64(Float64(sqrt(Float64(A + Float64(C + hypot(Float64(A - C), B_m)))) * Float64(-sqrt(Float64(2.0 * Float64(F * fma(B_m, B_m, Float64(A * Float64(C * -4.0)))))))) / 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(Float64(-sqrt(2.0)) / 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], 5e+99], N[(N[(N[Sqrt[N[(A + N[(C + N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * N[(F * N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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]) / B$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B 2) < 5.00000000000000008e99

   1. Initial program 25.6%

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

      1. sqrt-prod 27.4%

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

      2. associate-*r* 27.4%

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

      3. associate-*l* 27.4%

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

      4. associate-+l+ 28.1%

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

      5. unpow2 28.1%

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

      6. unpow2 28.1%

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

      7. hypot-def 37.5%

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

   3. Applied egg-rr 37.5%

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

      1. *-commutative 37.5%

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

      2. associate-*l* 37.5%

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

      3. *-commutative 37.5%

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

      4. unpow2 37.5%

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

      5. fma-neg 37.5%

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

      6. distribute-lft-neg-in 37.5%

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

      7. metadata-eval 37.5%

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

      8. *-commutative 37.5%

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

      9. associate-*l* 37.5%

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

   5. Simplified 37.5%

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

   if 5.00000000000000008e99 < (pow.f64 B 2)

   1. Initial program 6.2%

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

   2. Taylor expanded in A around 0 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 25.5%

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

   4. Simplified 25.5%

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

      1. pow1/2 25.5%

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

      2. *-commutative 25.5%

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

      3. unpow-prod-down 41.4%

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

      4. pow1/2 41.4%

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

      5. pow1/2 41.4%

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

   6. Applied egg-rr 41.4%

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

4. Final simplification 38.9%

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

Alternative 2: 40.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := {B_m}^{2} - C \cdot \left(A \cdot 4\right)\\ t_1 := \frac{-\sqrt{\left(2 \cdot \left(F \cdot t_0\right)\right) \cdot \left(2 \cdot C\right)}}{t_0}\\ \mathbf{if}\;{B_m}^{2} \leq 5 \cdot 10^{-322}:\\ \;\;\;\;\sqrt{-\frac{F}{C}}\\ \mathbf{elif}\;{B_m}^{2} \leq 2 \cdot 10^{-175}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;{B_m}^{2} \leq 10^{-100}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \left(-\sqrt{F \cdot \left(A + \mathsf{hypot}\left(B_m, A\right)\right)}\right)}{B_m}\\ \mathbf{elif}\;{B_m}^{2} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B_m} \cdot \left(\sqrt{B_m + C} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \end{array} \]

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))))
        (t_1 (/ (- (sqrt (* (* 2.0 (* F t_0)) (* 2.0 C)))) t_0)))
   (if (<= (pow B_m 2.0) 5e-322)
     (sqrt (- (/ F C)))
     (if (<= (pow B_m 2.0) 2e-175)
       t_1
       (if (<= (pow B_m 2.0) 1e-100)
         (/ (* (sqrt 2.0) (- (sqrt (* F (+ A (hypot B_m A)))))) B_m)
         (if (<= (pow B_m 2.0) 5e-21)
           t_1
           (* (/ (sqrt 2.0) B_m) (* (sqrt (+ B_m C)) (- (sqrt F))))))))))

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 t_1 = -sqrt(((2.0 * (F * t_0)) * (2.0 * C))) / t_0;
	double tmp;
	if (pow(B_m, 2.0) <= 5e-322) {
		tmp = sqrt(-(F / C));
	} else if (pow(B_m, 2.0) <= 2e-175) {
		tmp = t_1;
	} else if (pow(B_m, 2.0) <= 1e-100) {
		tmp = (sqrt(2.0) * -sqrt((F * (A + hypot(B_m, A))))) / B_m;
	} else if (pow(B_m, 2.0) <= 5e-21) {
		tmp = t_1;
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt((B_m + C)) * -sqrt(F));
	}
	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 t_1 = -Math.sqrt(((2.0 * (F * t_0)) * (2.0 * C))) / t_0;
	double tmp;
	if (Math.pow(B_m, 2.0) <= 5e-322) {
		tmp = Math.sqrt(-(F / C));
	} else if (Math.pow(B_m, 2.0) <= 2e-175) {
		tmp = t_1;
	} else if (Math.pow(B_m, 2.0) <= 1e-100) {
		tmp = (Math.sqrt(2.0) * -Math.sqrt((F * (A + Math.hypot(B_m, A))))) / B_m;
	} else if (Math.pow(B_m, 2.0) <= 5e-21) {
		tmp = t_1;
	} else {
		tmp = (Math.sqrt(2.0) / B_m) * (Math.sqrt((B_m + C)) * -Math.sqrt(F));
	}
	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))
	t_1 = -math.sqrt(((2.0 * (F * t_0)) * (2.0 * C))) / t_0
	tmp = 0
	if math.pow(B_m, 2.0) <= 5e-322:
		tmp = math.sqrt(-(F / C))
	elif math.pow(B_m, 2.0) <= 2e-175:
		tmp = t_1
	elif math.pow(B_m, 2.0) <= 1e-100:
		tmp = (math.sqrt(2.0) * -math.sqrt((F * (A + math.hypot(B_m, A))))) / B_m
	elif math.pow(B_m, 2.0) <= 5e-21:
		tmp = t_1
	else:
		tmp = (math.sqrt(2.0) / B_m) * (math.sqrt((B_m + C)) * -math.sqrt(F))
	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)))
	t_1 = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(F * t_0)) * Float64(2.0 * C)))) / t_0)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e-322)
		tmp = sqrt(Float64(-Float64(F / C)));
	elseif ((B_m ^ 2.0) <= 2e-175)
		tmp = t_1;
	elseif ((B_m ^ 2.0) <= 1e-100)
		tmp = Float64(Float64(sqrt(2.0) * Float64(-sqrt(Float64(F * Float64(A + hypot(B_m, A)))))) / B_m);
	elseif ((B_m ^ 2.0) <= 5e-21)
		tmp = t_1;
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(Float64(B_m + C)) * Float64(-sqrt(F))));
	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));
	t_1 = -sqrt(((2.0 * (F * t_0)) * (2.0 * C))) / t_0;
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 5e-322)
		tmp = sqrt(-(F / C));
	elseif ((B_m ^ 2.0) <= 2e-175)
		tmp = t_1;
	elseif ((B_m ^ 2.0) <= 1e-100)
		tmp = (sqrt(2.0) * -sqrt((F * (A + hypot(B_m, A))))) / B_m;
	elseif ((B_m ^ 2.0) <= 5e-21)
		tmp = t_1;
	else
		tmp = (sqrt(2.0) / B_m) * (sqrt((B_m + C)) * -sqrt(F));
	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]}, Block[{t$95$1 = N[((-N[Sqrt[N[(N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-322], N[Sqrt[(-N[(F / C), $MachinePrecision])], $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-175], t$95$1, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-100], 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], 5e-21], t$95$1, N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (pow.f64 B 2) < 4.99006e-322

   1. Initial program 17.2%

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

   3. Simplified 29.2%

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

      1. add-sqr-sqrt 12.2%

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

      2. sqrt-unprod 9.1%

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

      3. frac-times 8.9%

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

   5. Applied egg-rr 8.9%

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

   6. Taylor expanded in A around inf 27.3%

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

      1. mul-1-neg 27.3%

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

   8. Simplified 27.3%

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

   if 4.99006e-322 < (pow.f64 B 2) < 2e-175 or 1e-100 < (pow.f64 B 2) < 4.99999999999999973e-21

   1. Initial program 29.7%

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

   2. Taylor expanded in A around -inf 30.6%

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

   if 2e-175 < (pow.f64 B 2) < 1e-100

   1. Initial program 38.2%

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

   2. Taylor expanded in C around 0 19.6%

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

      1. mul-1-neg 19.6%

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

      2. *-commutative 19.6%

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

      3. distribute-rgt-neg-in 19.6%

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

      4. +-commutative 19.6%

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

      5. unpow2 19.6%

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

      6. unpow2 19.6%

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

      7. hypot-def 20.0%

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

   4. Simplified 20.0%

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

      1. pow1/2 20.0%

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

      2. *-commutative 20.0%

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

      3. unpow-prod-down 20.1%

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

      4. pow1/2 20.1%

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

      5. pow1/2 20.1%

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

   6. Applied egg-rr 20.1%

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

      1. distribute-neg-frac 20.1%

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

      2. associate-*r/ 20.2%

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

      3. *-commutative 20.2%

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

      4. sqrt-unprod 20.2%

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

   8. Applied egg-rr 20.2%

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

   if 4.99999999999999973e-21 < (pow.f64 B 2)

   1. Initial program 10.9%

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

   2. Taylor expanded in A around 0 10.7%

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

      1. mul-1-neg 10.7%

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

      2. *-commutative 10.7%

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

      3. distribute-rgt-neg-in 10.7%

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

      4. unpow2 10.7%

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

      5. unpow2 10.7%

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

      6. hypot-def 23.0%

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

   4. Simplified 23.0%

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

      1. pow1/2 23.0%

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

      2. *-commutative 23.0%

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

      3. unpow-prod-down 36.9%

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

      4. pow1/2 36.9%

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

      5. pow1/2 36.9%

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

   6. Applied egg-rr 36.9%

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

   7. Taylor expanded in C around 0 31.2%

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

4. Final simplification 29.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-322}:\\ \;\;\;\;\sqrt{-\frac{F}{C}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{-175}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right)\right) \cdot \left(2 \cdot C\right)}}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;{B}^{2} \leq 10^{-100}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \left(-\sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)}\right)}{B}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right)\right) \cdot \left(2 \cdot C\right)}}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{B + C} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]

Alternative 3: 44.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B_m, B_m, -4 \cdot \left(A \cdot C\right)\right)\\ t_1 := {B_m}^{2} - C \cdot \left(A \cdot 4\right)\\ \mathbf{if}\;{B_m}^{2} \leq 5 \cdot 10^{-322}:\\ \;\;\;\;\sqrt{-\frac{F}{C}}\\ \mathbf{elif}\;{B_m}^{2} \leq 2 \cdot 10^{-175}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot t_1\right)\right) \cdot \left(2 \cdot C\right)}}{t_1}\\ \mathbf{elif}\;{B_m}^{2} \leq 4 \cdot 10^{-58}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot t_0\right) \cdot \left(F \cdot \left(A + \mathsf{hypot}\left(B_m, A\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(B_m, C\right)} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{2}}{B_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* -4.0 (* A C))))
        (t_1 (- (pow B_m 2.0) (* C (* A 4.0)))))
   (if (<= (pow B_m 2.0) 5e-322)
     (sqrt (- (/ F C)))
     (if (<= (pow B_m 2.0) 2e-175)
       (/ (- (sqrt (* (* 2.0 (* F t_1)) (* 2.0 C)))) t_1)
       (if (<= (pow B_m 2.0) 4e-58)
         (/ (- (sqrt (* (* 2.0 t_0) (* F (+ A (hypot B_m A)))))) t_0)
         (*
          (* (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 t_0 = fma(B_m, B_m, (-4.0 * (A * C)));
	double t_1 = pow(B_m, 2.0) - (C * (A * 4.0));
	double tmp;
	if (pow(B_m, 2.0) <= 5e-322) {
		tmp = sqrt(-(F / C));
	} else if (pow(B_m, 2.0) <= 2e-175) {
		tmp = -sqrt(((2.0 * (F * t_1)) * (2.0 * C))) / t_1;
	} else if (pow(B_m, 2.0) <= 4e-58) {
		tmp = -sqrt(((2.0 * t_0) * (F * (A + hypot(B_m, A))))) / t_0;
	} else {
		tmp = (sqrt((C + hypot(B_m, C))) * sqrt(F)) * (-sqrt(2.0) / B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(-4.0 * Float64(A * C)))
	t_1 = Float64((B_m ^ 2.0) - Float64(C * Float64(A * 4.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e-322)
		tmp = sqrt(Float64(-Float64(F / C)));
	elseif ((B_m ^ 2.0) <= 2e-175)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(F * t_1)) * Float64(2.0 * C)))) / t_1);
	elseif ((B_m ^ 2.0) <= 4e-58)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * t_0) * Float64(F * Float64(A + hypot(B_m, A)))))) / t_0);
	else
		tmp = Float64(Float64(sqrt(Float64(C + hypot(B_m, C))) * sqrt(F)) * Float64(Float64(-sqrt(2.0)) / 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[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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], 5e-322], N[Sqrt[(-N[(F / C), $MachinePrecision])], $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-175], N[((-N[Sqrt[N[(N[(2.0 * N[(F * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e-58], N[((-N[Sqrt[N[(N[(2.0 * t$95$0), $MachinePrecision] * N[(F * N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 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]) / B$95$m), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (pow.f64 B 2) < 4.99006e-322

   1. Initial program 17.2%

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

   3. Simplified 29.2%

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

      1. add-sqr-sqrt 12.2%

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

      2. sqrt-unprod 9.1%

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

      3. frac-times 8.9%

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

   5. Applied egg-rr 8.9%

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

   6. Taylor expanded in A around inf 27.3%

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

      1. mul-1-neg 27.3%

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

   8. Simplified 27.3%

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

   if 4.99006e-322 < (pow.f64 B 2) < 2e-175

   1. Initial program 26.9%

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

   2. Taylor expanded in A around -inf 29.5%

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

   if 2e-175 < (pow.f64 B 2) < 4.0000000000000001e-58

   1. Initial program 35.4%

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

      1. neg-sub0 35.4%

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

      2. div-sub 35.4%

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

      3. associate-*l* 35.4%

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

   3. Applied egg-rr 44.4%

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

      1. div0 44.4%

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

      2. neg-sub0 44.4%

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

      3. distribute-neg-frac 44.4%

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

   5. Simplified 42.7%

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

   6. Taylor expanded in C around 0 32.5%

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

      1. +-commutative 32.5%

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

      2. unpow2 32.5%

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

      3. unpow2 32.5%

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

      4. hypot-def 36.5%

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

   8. Simplified 36.5%

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

   if 4.0000000000000001e-58 < (pow.f64 B 2)

   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. Taylor expanded in A around 0 10.7%

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

      1. mul-1-neg 10.7%

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

      2. *-commutative 10.7%

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

      3. distribute-rgt-neg-in 10.7%

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

      4. unpow2 10.7%

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

      5. unpow2 10.7%

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

      6. hypot-def 21.8%

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

   4. Simplified 21.8%

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

      1. pow1/2 21.8%

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

      2. *-commutative 21.8%

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

      3. unpow-prod-down 34.3%

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

      4. pow1/2 34.3%

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

      5. pow1/2 34.3%

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

   6. Applied egg-rr 34.3%

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

4. Final simplification 32.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-322}:\\ \;\;\;\;\sqrt{-\frac{F}{C}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{-175}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right)\right) \cdot \left(2 \cdot C\right)}}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;{B}^{2} \leq 4 \cdot 10^{-58}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)\right) \cdot \left(F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)\right)}}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 4: 43.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := {B_m}^{2} - C \cdot \left(A \cdot 4\right)\\ \mathbf{if}\;{B_m}^{2} \leq 5 \cdot 10^{-322}:\\ \;\;\;\;\sqrt{-\frac{F}{C}}\\ \mathbf{elif}\;{B_m}^{2} \leq 2 \cdot 10^{-175}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot t_1\right)\right) \cdot \left(2 \cdot C\right)}}{t_1}\\ \mathbf{elif}\;{B_m}^{2} \leq 5 \cdot 10^{-75}:\\ \;\;\;\;\frac{-\sqrt{\left(t_0 \cdot \left(2 \cdot F\right)\right) \cdot \left(A + A\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(B_m, C\right)} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{2}}{B_m}\\ \end{array} \end{array} \]

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 (- (pow B_m 2.0) (* C (* A 4.0)))))
   (if (<= (pow B_m 2.0) 5e-322)
     (sqrt (- (/ F C)))
     (if (<= (pow B_m 2.0) 2e-175)
       (/ (- (sqrt (* (* 2.0 (* F t_1)) (* 2.0 C)))) t_1)
       (if (<= (pow B_m 2.0) 5e-75)
         (/ (- (sqrt (* (* t_0 (* 2.0 F)) (+ A A)))) t_0)
         (*
          (* (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 t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_1 = pow(B_m, 2.0) - (C * (A * 4.0));
	double tmp;
	if (pow(B_m, 2.0) <= 5e-322) {
		tmp = sqrt(-(F / C));
	} else if (pow(B_m, 2.0) <= 2e-175) {
		tmp = -sqrt(((2.0 * (F * t_1)) * (2.0 * C))) / t_1;
	} else if (pow(B_m, 2.0) <= 5e-75) {
		tmp = -sqrt(((t_0 * (2.0 * F)) * (A + A))) / t_0;
	} else {
		tmp = (sqrt((C + hypot(B_m, C))) * sqrt(F)) * (-sqrt(2.0) / 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)))
	t_1 = Float64((B_m ^ 2.0) - Float64(C * Float64(A * 4.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e-322)
		tmp = sqrt(Float64(-Float64(F / C)));
	elseif ((B_m ^ 2.0) <= 2e-175)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(F * t_1)) * Float64(2.0 * C)))) / t_1);
	elseif ((B_m ^ 2.0) <= 5e-75)
		tmp = Float64(Float64(-sqrt(Float64(Float64(t_0 * Float64(2.0 * F)) * Float64(A + A)))) / t_0);
	else
		tmp = Float64(Float64(sqrt(Float64(C + hypot(B_m, C))) * sqrt(F)) * Float64(Float64(-sqrt(2.0)) / 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]}, Block[{t$95$1 = 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], 5e-322], N[Sqrt[(-N[(F / C), $MachinePrecision])], $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-175], N[((-N[Sqrt[N[(N[(2.0 * N[(F * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-75], N[((-N[Sqrt[N[(N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision] * N[(A + A), $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]) / B$95$m), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (pow.f64 B 2) < 4.99006e-322

   1. Initial program 17.2%

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

   3. Simplified 29.2%

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

      1. add-sqr-sqrt 12.2%

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

      2. sqrt-unprod 9.1%

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

      3. frac-times 8.9%

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

   5. Applied egg-rr 8.9%

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

   6. Taylor expanded in A around inf 27.3%

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

      1. mul-1-neg 27.3%

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

   8. Simplified 27.3%

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

   if 4.99006e-322 < (pow.f64 B 2) < 2e-175

   1. Initial program 26.9%

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

   2. Taylor expanded in A around -inf 29.5%

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

   if 2e-175 < (pow.f64 B 2) < 4.99999999999999979e-75

   1. Initial program 28.2%

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

   3. Simplified 35.4%

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

   4. Taylor expanded in A around inf 33.1%

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

      1. distribute-rgt1-in 33.1%

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

      2. metadata-eval 33.1%

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

      3. mul0-lft 33.1%

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

   6. Simplified 33.1%

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

   if 4.99999999999999979e-75 < (pow.f64 B 2)

   1. Initial program 15.6%

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

   2. Taylor expanded in A around 0 10.3%

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

      1. mul-1-neg 10.3%

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

      2. *-commutative 10.3%

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

      3. distribute-rgt-neg-in 10.3%

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

      4. unpow2 10.3%

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

      5. unpow2 10.3%

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

      6. hypot-def 20.8%

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

   4. Simplified 20.8%

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

      1. pow1/2 20.8%

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

      2. *-commutative 20.8%

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

      3. unpow-prod-down 32.7%

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

      4. pow1/2 32.7%

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

      5. pow1/2 32.7%

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

   6. Applied egg-rr 32.7%

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

4. Final simplification 31.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-322}:\\ \;\;\;\;\sqrt{-\frac{F}{C}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{-175}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right)\right) \cdot \left(2 \cdot C\right)}}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{-75}:\\ \;\;\;\;\frac{-\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(A + A\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 5: 40.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := {B_m}^{2} - C \cdot \left(A \cdot 4\right)\\ \mathbf{if}\;{B_m}^{2} \leq 5 \cdot 10^{-322}:\\ \;\;\;\;\sqrt{-\frac{F}{C}}\\ \mathbf{elif}\;{B_m}^{2} \leq 2 \cdot 10^{-175}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot t_1\right)\right) \cdot \left(2 \cdot C\right)}}{t_1}\\ \mathbf{elif}\;{B_m}^{2} \leq 5 \cdot 10^{-75}:\\ \;\;\;\;\frac{-\sqrt{\left(t_0 \cdot \left(2 \cdot F\right)\right) \cdot \left(A + A\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B_m} \cdot \left(\sqrt{B_m + C} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \end{array} \]

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 (- (pow B_m 2.0) (* C (* A 4.0)))))
   (if (<= (pow B_m 2.0) 5e-322)
     (sqrt (- (/ F C)))
     (if (<= (pow B_m 2.0) 2e-175)
       (/ (- (sqrt (* (* 2.0 (* F t_1)) (* 2.0 C)))) t_1)
       (if (<= (pow B_m 2.0) 5e-75)
         (/ (- (sqrt (* (* t_0 (* 2.0 F)) (+ A A)))) t_0)
         (* (/ (sqrt 2.0) B_m) (* (sqrt (+ B_m C)) (- (sqrt F)))))))))

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 = pow(B_m, 2.0) - (C * (A * 4.0));
	double tmp;
	if (pow(B_m, 2.0) <= 5e-322) {
		tmp = sqrt(-(F / C));
	} else if (pow(B_m, 2.0) <= 2e-175) {
		tmp = -sqrt(((2.0 * (F * t_1)) * (2.0 * C))) / t_1;
	} else if (pow(B_m, 2.0) <= 5e-75) {
		tmp = -sqrt(((t_0 * (2.0 * F)) * (A + A))) / t_0;
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt((B_m + C)) * -sqrt(F));
	}
	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((B_m ^ 2.0) - Float64(C * Float64(A * 4.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e-322)
		tmp = sqrt(Float64(-Float64(F / C)));
	elseif ((B_m ^ 2.0) <= 2e-175)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(F * t_1)) * Float64(2.0 * C)))) / t_1);
	elseif ((B_m ^ 2.0) <= 5e-75)
		tmp = Float64(Float64(-sqrt(Float64(Float64(t_0 * Float64(2.0 * F)) * Float64(A + A)))) / t_0);
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(Float64(B_m + C)) * Float64(-sqrt(F))));
	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[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], 5e-322], N[Sqrt[(-N[(F / C), $MachinePrecision])], $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-175], N[((-N[Sqrt[N[(N[(2.0 * N[(F * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-75], N[((-N[Sqrt[N[(N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision] * N[(A + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (pow.f64 B 2) < 4.99006e-322

   1. Initial program 17.2%

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

   3. Simplified 29.2%

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

      1. add-sqr-sqrt 12.2%

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

      2. sqrt-unprod 9.1%

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

      3. frac-times 8.9%

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

   5. Applied egg-rr 8.9%

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

   6. Taylor expanded in A around inf 27.3%

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

      1. mul-1-neg 27.3%

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

   8. Simplified 27.3%

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

   if 4.99006e-322 < (pow.f64 B 2) < 2e-175

   1. Initial program 26.9%

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

   2. Taylor expanded in A around -inf 29.5%

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

   if 2e-175 < (pow.f64 B 2) < 4.99999999999999979e-75

   1. Initial program 28.2%

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

   3. Simplified 35.4%

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

   4. Taylor expanded in A around inf 33.1%

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

      1. distribute-rgt1-in 33.1%

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

      2. metadata-eval 33.1%

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

      3. mul0-lft 33.1%

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

   6. Simplified 33.1%

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

   if 4.99999999999999979e-75 < (pow.f64 B 2)

   1. Initial program 15.6%

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

   2. Taylor expanded in A around 0 10.3%

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

      1. mul-1-neg 10.3%

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

      2. *-commutative 10.3%

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

      3. distribute-rgt-neg-in 10.3%

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

      4. unpow2 10.3%

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

      5. unpow2 10.3%

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

      6. hypot-def 20.8%

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

   4. Simplified 20.8%

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

      1. pow1/2 20.8%

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

      2. *-commutative 20.8%

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

      3. unpow-prod-down 32.7%

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

      4. pow1/2 32.7%

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

      5. pow1/2 32.7%

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

   6. Applied egg-rr 32.7%

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

   7. Taylor expanded in C around 0 27.7%

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

4. Final simplification 28.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-322}:\\ \;\;\;\;\sqrt{-\frac{F}{C}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{-175}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right)\right) \cdot \left(2 \cdot C\right)}}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{-75}:\\ \;\;\;\;\frac{-\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(A + A\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{B + C} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]

Alternative 6: 51.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;{B_m}^{2} \leq 5 \cdot 10^{+99}:\\ \;\;\;\;\frac{\sqrt{\mathsf{hypot}\left(A - C, B_m\right) + \left(A + C\right)} \cdot \left(-\sqrt{\mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)}\right)}{\mathsf{fma}\left(B_m, B_m, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(B_m, C\right)} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{2}}{B_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 5e+99)
   (/
    (*
     (sqrt (+ (hypot (- A C) B_m) (+ A C)))
     (- (sqrt (* (fma B_m B_m (* A (* C -4.0))) (* 2.0 F)))))
    (fma B_m B_m (* -4.0 (* A 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 (pow(B_m, 2.0) <= 5e+99) {
		tmp = (sqrt((hypot((A - C), B_m) + (A + C))) * -sqrt((fma(B_m, B_m, (A * (C * -4.0))) * (2.0 * F)))) / fma(B_m, B_m, (-4.0 * (A * C)));
	} else {
		tmp = (sqrt((C + hypot(B_m, C))) * sqrt(F)) * (-sqrt(2.0) / B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e+99)
		tmp = Float64(Float64(sqrt(Float64(hypot(Float64(A - C), B_m) + Float64(A + C))) * Float64(-sqrt(Float64(fma(B_m, B_m, Float64(A * Float64(C * -4.0))) * Float64(2.0 * F))))) / fma(B_m, B_m, Float64(-4.0 * Float64(A * 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

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+99], N[(N[(N[Sqrt[N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + N[(A + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $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]) / B$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B 2) < 5.00000000000000008e99

   1. Initial program 25.6%

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

      1. neg-sub0 25.6%

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

      2. div-sub 25.6%

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

      3. associate-*l* 25.6%

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

   3. Applied egg-rr 33.6%

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

      1. div0 33.6%

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

      2. neg-sub0 33.6%

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

      3. distribute-neg-frac 33.6%

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

   5. Simplified 31.6%

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

      1. pow1/2 31.6%

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

      2. associate-*r* 31.9%

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

      3. *-commutative 31.9%

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

      4. *-commutative 31.9%

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

      5. associate-*r* 31.9%

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

      6. +-commutative 31.9%

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

      7. associate-+r+ 33.6%

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

      8. *-commutative 33.6%

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

      9. unpow-prod-down 37.5%

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

      10. pow1/2 37.5%

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

      11. associate-+r+ 36.3%

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

      12. +-commutative 36.3%

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

      13. pow1/2 36.3%

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

   7. Applied egg-rr 36.3%

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

   if 5.00000000000000008e99 < (pow.f64 B 2)

   1. Initial program 6.2%

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

   2. Taylor expanded in A around 0 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 25.5%

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

   4. Simplified 25.5%

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

      1. pow1/2 25.5%

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

      2. *-commutative 25.5%

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

      3. unpow-prod-down 41.4%

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

      4. pow1/2 41.4%

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

      5. pow1/2 41.4%

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

   6. Applied egg-rr 41.4%

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

4. Final simplification 38.1%

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

Alternative 7: 44.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := {B_m}^{2} - C \cdot \left(A \cdot 4\right)\\ t_1 := 2 \cdot \left(F \cdot t_0\right)\\ \mathbf{if}\;B_m \leq 6.8 \cdot 10^{-161}:\\ \;\;\;\;\sqrt{-\frac{F}{C}}\\ \mathbf{elif}\;B_m \leq 6.8 \cdot 10^{-84}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(2 \cdot C\right)}}{t_0}\\ \mathbf{elif}\;B_m \leq 2 \cdot 10^{-29}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(A + \mathsf{hypot}\left(B_m, A\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(B_m, C\right)} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{2}}{B_m}\\ \end{array} \end{array} \]

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)))) (t_1 (* 2.0 (* F t_0))))
   (if (<= B_m 6.8e-161)
     (sqrt (- (/ F C)))
     (if (<= B_m 6.8e-84)
       (/ (- (sqrt (* t_1 (* 2.0 C)))) t_0)
       (if (<= B_m 2e-29)
         (/ (- (sqrt (* t_1 (+ A (hypot B_m A))))) t_0)
         (*
          (* (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 t_0 = pow(B_m, 2.0) - (C * (A * 4.0));
	double t_1 = 2.0 * (F * t_0);
	double tmp;
	if (B_m <= 6.8e-161) {
		tmp = sqrt(-(F / C));
	} else if (B_m <= 6.8e-84) {
		tmp = -sqrt((t_1 * (2.0 * C))) / t_0;
	} else if (B_m <= 2e-29) {
		tmp = -sqrt((t_1 * (A + hypot(B_m, A)))) / t_0;
	} 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 t_0 = Math.pow(B_m, 2.0) - (C * (A * 4.0));
	double t_1 = 2.0 * (F * t_0);
	double tmp;
	if (B_m <= 6.8e-161) {
		tmp = Math.sqrt(-(F / C));
	} else if (B_m <= 6.8e-84) {
		tmp = -Math.sqrt((t_1 * (2.0 * C))) / t_0;
	} else if (B_m <= 2e-29) {
		tmp = -Math.sqrt((t_1 * (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) / 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))
	t_1 = 2.0 * (F * t_0)
	tmp = 0
	if B_m <= 6.8e-161:
		tmp = math.sqrt(-(F / C))
	elif B_m <= 6.8e-84:
		tmp = -math.sqrt((t_1 * (2.0 * C))) / t_0
	elif B_m <= 2e-29:
		tmp = -math.sqrt((t_1 * (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) / 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)))
	t_1 = Float64(2.0 * Float64(F * t_0))
	tmp = 0.0
	if (B_m <= 6.8e-161)
		tmp = sqrt(Float64(-Float64(F / C)));
	elseif (B_m <= 6.8e-84)
		tmp = Float64(Float64(-sqrt(Float64(t_1 * Float64(2.0 * C)))) / t_0);
	elseif (B_m <= 2e-29)
		tmp = Float64(Float64(-sqrt(Float64(t_1 * Float64(A + hypot(B_m, A))))) / t_0);
	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)
	t_0 = (B_m ^ 2.0) - (C * (A * 4.0));
	t_1 = 2.0 * (F * t_0);
	tmp = 0.0;
	if (B_m <= 6.8e-161)
		tmp = sqrt(-(F / C));
	elseif (B_m <= 6.8e-84)
		tmp = -sqrt((t_1 * (2.0 * C))) / t_0;
	elseif (B_m <= 2e-29)
		tmp = -sqrt((t_1 * (A + hypot(B_m, A)))) / t_0;
	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_] := Block[{t$95$0 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 6.8e-161], N[Sqrt[(-N[(F / C), $MachinePrecision])], $MachinePrecision], If[LessEqual[B$95$m, 6.8e-84], N[((-N[Sqrt[N[(t$95$1 * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[B$95$m, 2e-29], N[((-N[Sqrt[N[(t$95$1 * 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]) / B$95$m), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 6.79999999999999964e-161

   1. Initial program 18.7%

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

   3. Simplified 25.5%

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

      1. add-sqr-sqrt 6.0%

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

      2. sqrt-unprod 5.3%

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

      3. frac-times 5.0%

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

   5. Applied egg-rr 5.0%

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

   6. Taylor expanded in A around inf 14.3%

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

      1. mul-1-neg 14.3%

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

   8. Simplified 14.3%

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

   if 6.79999999999999964e-161 < B < 6.80000000000000042e-84

   1. Initial program 29.4%

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

   2. Taylor expanded in A around -inf 35.0%

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

   if 6.80000000000000042e-84 < B < 1.99999999999999989e-29

   1. Initial program 33.6%

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

   2. Taylor expanded in C around 0 23.3%

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

      1. +-commutative 23.3%

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

      2. unpow2 23.3%

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

      3. unpow2 23.3%

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

      4. hypot-def 24.9%

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

   4. Simplified 24.9%

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

   if 1.99999999999999989e-29 < B

   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 19.9%

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

      1. mul-1-neg 19.9%

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

      2. *-commutative 19.9%

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

      3. distribute-rgt-neg-in 19.9%

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

      4. unpow2 19.9%

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

      5. unpow2 19.9%

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

      6. hypot-def 41.1%

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

   4. Simplified 41.1%

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

      1. pow1/2 41.1%

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

      2. *-commutative 41.1%

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

      3. unpow-prod-down 65.8%

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

      4. pow1/2 65.8%

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

      5. pow1/2 65.8%

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

   6. Applied egg-rr 65.8%

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

4. Final simplification 29.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 6.8 \cdot 10^{-161}:\\ \;\;\;\;\sqrt{-\frac{F}{C}}\\ \mathbf{elif}\;B \leq 6.8 \cdot 10^{-84}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right)\right) \cdot \left(2 \cdot C\right)}}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;B \leq 2 \cdot 10^{-29}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right)\right) \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)}}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 8: 47.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B_m, B_m, -4 \cdot \left(A \cdot C\right)\right)\\ \mathbf{if}\;B_m \leq 7.4 \cdot 10^{+28}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot t_0\right) \cdot \left(F \cdot \left(\mathsf{hypot}\left(A - C, B_m\right) + \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(B_m, C\right)} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{2}}{B_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* -4.0 (* A C)))))
   (if (<= B_m 7.4e+28)
     (/ (- (sqrt (* (* 2.0 t_0) (* F (+ (hypot (- A C) B_m) (+ A C)))))) t_0)
     (* (* (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 t_0 = fma(B_m, B_m, (-4.0 * (A * C)));
	double tmp;
	if (B_m <= 7.4e+28) {
		tmp = -sqrt(((2.0 * t_0) * (F * (hypot((A - C), B_m) + (A + C))))) / t_0;
	} else {
		tmp = (sqrt((C + hypot(B_m, C))) * sqrt(F)) * (-sqrt(2.0) / B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(-4.0 * Float64(A * C)))
	tmp = 0.0
	if (B_m <= 7.4e+28)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * t_0) * Float64(F * Float64(hypot(Float64(A - C), B_m) + Float64(A + C)))))) / t_0);
	else
		tmp = Float64(Float64(sqrt(Float64(C + hypot(B_m, C))) * sqrt(F)) * Float64(Float64(-sqrt(2.0)) / 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[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 7.4e+28], N[((-N[Sqrt[N[(N[(2.0 * t$95$0), $MachinePrecision] * N[(F * N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + N[(A + C), $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]) / B$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 7.3999999999999998e28

   1. Initial program 21.0%

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

      1. neg-sub0 21.0%

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

      2. div-sub 21.0%

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

      3. associate-*l* 21.0%

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

   3. Applied egg-rr 27.4%

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

      1. div0 27.4%

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

      2. neg-sub0 27.4%

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

      3. distribute-neg-frac 27.4%

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

   5. Simplified 25.8%

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

   if 7.3999999999999998e28 < B

   1. Initial program 8.9%

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

   2. Taylor expanded in A around 0 19.0%

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

      1. mul-1-neg 19.0%

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

      2. *-commutative 19.0%

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

      3. distribute-rgt-neg-in 19.0%

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

      4. unpow2 19.0%

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

      5. unpow2 19.0%

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

      6. hypot-def 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 78.3%

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

      4. pow1/2 78.3%

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

      5. pow1/2 78.3%

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

   6. Applied egg-rr 78.3%

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

4. Final simplification 36.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 7.4 \cdot 10^{+28}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)\right) \cdot \left(F \cdot \left(\mathsf{hypot}\left(A - C, B\right) + \left(A + C\right)\right)\right)}}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 9: 48.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B_m \leq 7.8 \cdot 10^{+28}:\\ \;\;\;\;\frac{-\sqrt{\left(t_0 \cdot \left(2 \cdot F\right)\right) \cdot \left(A + \left(C + \mathsf{hypot}\left(B_m, A - C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(B_m, C\right)} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{2}}{B_m}\\ \end{array} \end{array} \]

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 (<= B_m 7.8e+28)
     (/ (- (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)) 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 (B_m <= 7.8e+28) {
		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) / 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 <= 7.8e+28)
		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(Float64(-sqrt(2.0)) / 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[B$95$m, 7.8e+28], 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]) / B$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 7.7999999999999997e28

   1. Initial program 21.0%

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

   3. Simplified 27.4%

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

   if 7.7999999999999997e28 < B

   1. Initial program 8.9%

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

   2. Taylor expanded in A around 0 19.0%

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

      1. mul-1-neg 19.0%

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

      2. *-commutative 19.0%

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

      3. distribute-rgt-neg-in 19.0%

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

      4. unpow2 19.0%

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

      5. unpow2 19.0%

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

      6. hypot-def 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 78.3%

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

      4. pow1/2 78.3%

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

      5. pow1/2 78.3%

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

   6. Applied egg-rr 78.3%

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

4. Final simplification 37.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 7.8 \cdot 10^{+28}:\\ \;\;\;\;\frac{-\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 10: 46.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{\sqrt{-F}}{\sqrt{A}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B_m} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{A + \mathsf{hypot}\left(B_m, A\right)}\right)\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F -5e-311)
   (/ (sqrt (- F)) (sqrt A))
   (* (/ (sqrt 2.0) B_m) (* (sqrt F) (- (sqrt (+ A (hypot B_m A))))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= -5e-311) {
		tmp = sqrt(-F) / sqrt(A);
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt(F) * -sqrt((A + hypot(B_m, A))));
	}
	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 <= -5e-311) {
		tmp = Math.sqrt(-F) / Math.sqrt(A);
	} else {
		tmp = (Math.sqrt(2.0) / B_m) * (Math.sqrt(F) * -Math.sqrt((A + Math.hypot(B_m, A))));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if F <= -5e-311:
		tmp = math.sqrt(-F) / math.sqrt(A)
	else:
		tmp = (math.sqrt(2.0) / B_m) * (math.sqrt(F) * -math.sqrt((A + math.hypot(B_m, A))))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= -5e-311)
		tmp = Float64(sqrt(Float64(-F)) / sqrt(A));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(F) * Float64(-sqrt(Float64(A + hypot(B_m, A))))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= -5e-311)
		tmp = sqrt(-F) / sqrt(A);
	else
		tmp = (sqrt(2.0) / B_m) * (sqrt(F) * -sqrt((A + hypot(B_m, A))));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, -5e-311], N[(N[Sqrt[(-F)], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -5.00000000000023e-311

   1. Initial program 37.7%

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

   3. Simplified 45.8%

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

      1. add-sqr-sqrt 45.7%

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

      2. sqrt-unprod 38.1%

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

      3. frac-times 33.8%

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

   5. Applied egg-rr 33.8%

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

   6. Taylor expanded in A around -inf 45.1%

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

      1. mul-1-neg 45.1%

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

   8. Simplified 45.1%

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

      1. distribute-neg-frac 45.1%

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

      2. sqrt-div 60.6%

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

   10. Applied egg-rr 60.6%

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

   if -5.00000000000023e-311 < F

   1. Initial program 16.7%

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

   2. Taylor expanded in C around 0 8.7%

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

      1. mul-1-neg 8.7%

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

      2. *-commutative 8.7%

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

      3. distribute-rgt-neg-in 8.7%

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

      4. +-commutative 8.7%

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

      5. unpow2 8.7%

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

      6. unpow2 8.7%

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

      7. hypot-def 15.4%

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

   4. Simplified 15.4%

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

      1. pow1/2 15.4%

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

      2. *-commutative 15.4%

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

      3. unpow-prod-down 21.5%

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

      4. pow1/2 21.5%

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

      5. pow1/2 21.5%

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

   6. Applied egg-rr 21.5%

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

4. Final simplification 25.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{\sqrt{-F}}{\sqrt{A}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{A + \mathsf{hypot}\left(B, A\right)}\right)\right)\\ \end{array} \]

Alternative 11: 43.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{\sqrt{-F}}{\sqrt{A}}\\ \mathbf{elif}\;F \leq 6.5 \cdot 10^{+23}:\\ \;\;\;\;\frac{-\sqrt{F \cdot \left(2 \cdot \left(C + \mathsf{hypot}\left(B_m, C\right)\right)\right)}}{B_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B_m} \cdot \left(\sqrt{F} \cdot \sqrt{B_m}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F -5e-311)
   (/ (sqrt (- F)) (sqrt A))
   (if (<= F 6.5e+23)
     (/ (- (sqrt (* F (* 2.0 (+ C (hypot B_m C)))))) B_m)
     (* (/ (- (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 <= -5e-311) {
		tmp = sqrt(-F) / sqrt(A);
	} else if (F <= 6.5e+23) {
		tmp = -sqrt((F * (2.0 * (C + hypot(B_m, C))))) / B_m;
	} else {
		tmp = (-sqrt(2.0) / B_m) * (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 tmp;
	if (F <= -5e-311) {
		tmp = Math.sqrt(-F) / Math.sqrt(A);
	} else if (F <= 6.5e+23) {
		tmp = -Math.sqrt((F * (2.0 * (C + Math.hypot(B_m, C))))) / B_m;
	} 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 <= -5e-311:
		tmp = math.sqrt(-F) / math.sqrt(A)
	elif F <= 6.5e+23:
		tmp = -math.sqrt((F * (2.0 * (C + math.hypot(B_m, C))))) / B_m
	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 <= -5e-311)
		tmp = Float64(sqrt(Float64(-F)) / sqrt(A));
	elseif (F <= 6.5e+23)
		tmp = Float64(Float64(-sqrt(Float64(F * Float64(2.0 * Float64(C + hypot(B_m, C)))))) / B_m);
	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 <= -5e-311)
		tmp = sqrt(-F) / sqrt(A);
	elseif (F <= 6.5e+23)
		tmp = -sqrt((F * (2.0 * (C + hypot(B_m, C))))) / B_m;
	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, -5e-311], N[(N[Sqrt[(-F)], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 6.5e+23], N[((-N[Sqrt[N[(F * N[(2.0 * N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / B$95$m), $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 3 regimes

2. if F < -5.00000000000023e-311

   1. Initial program 37.7%

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

   3. Simplified 45.8%

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

      1. add-sqr-sqrt 45.7%

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

      2. sqrt-unprod 38.1%

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

      3. frac-times 33.8%

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

   5. Applied egg-rr 33.8%

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

   6. Taylor expanded in A around -inf 45.1%

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

      1. mul-1-neg 45.1%

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

   8. Simplified 45.1%

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

      1. distribute-neg-frac 45.1%

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

      2. sqrt-div 60.6%

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

   10. Applied egg-rr 60.6%

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

   if -5.00000000000023e-311 < F < 6.4999999999999996e23

   1. Initial program 18.6%

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

   2. Taylor expanded in A around 0 9.0%

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

      1. mul-1-neg 9.0%

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

      2. *-commutative 9.0%

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

      3. distribute-rgt-neg-in 9.0%

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

      4. unpow2 9.0%

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

      5. unpow2 9.0%

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

      6. hypot-def 20.3%

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

   4. Simplified 20.3%

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

      1. pow1/2 20.3%

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

      2. *-commutative 20.3%

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

      3. unpow-prod-down 20.3%

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

      4. pow1/2 20.3%

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

      5. pow1/2 20.3%

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

   6. Applied egg-rr 20.3%

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

      1. *-commutative 20.3%

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

      2. sqrt-prod 20.3%

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

      3. distribute-rgt-neg-out 20.3%

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

      4. *-commutative 20.3%

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

      5. neg-sub0 20.3%

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

      6. associate-*l/ 20.2%

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

      7. sqrt-unprod 20.3%

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

      8. *-commutative 20.3%

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

   8. Applied egg-rr 20.3%

      \[\leadsto \color{blue}{0 - \frac{\sqrt{2 \cdot \left(\left(C + \mathsf{hypot}\left(B, C\right)\right) \cdot F\right)}}{B}} \]
   9. Step-by-step derivation

      1. neg-sub0 20.3%

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

      2. distribute-frac-neg 20.3%

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

      3. *-commutative 20.3%

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

      4. *-commutative 20.3%

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

      5. associate-*l* 20.3%

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

   10. Simplified 20.3%

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

   if 6.4999999999999996e23 < F

   1. Initial program 14.4%

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

   2. Taylor expanded in A around 0 8.1%

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

      1. mul-1-neg 8.1%

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

      2. *-commutative 8.1%

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

      3. distribute-rgt-neg-in 8.1%

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

      4. unpow2 8.1%

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

      5. unpow2 8.1%

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

      6. hypot-def 9.3%

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

   4. Simplified 9.3%

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

      1. pow1/2 9.3%

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

      2. *-commutative 9.3%

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

      3. unpow-prod-down 24.6%

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

      4. pow1/2 24.6%

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

      5. pow1/2 24.6%

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

   6. Applied egg-rr 24.6%

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

   7. Taylor expanded in C around 0 20.0%

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

4. Final simplification 24.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{\sqrt{-F}}{\sqrt{A}}\\ \mathbf{elif}\;F \leq 6.5 \cdot 10^{+23}:\\ \;\;\;\;\frac{-\sqrt{F \cdot \left(2 \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)\right)}}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \sqrt{B}\right)\\ \end{array} \]

Alternative 12: 42.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{\sqrt{-F}}{\sqrt{A}}\\ \mathbf{elif}\;F \leq 4.2 \cdot 10^{+25}:\\ \;\;\;\;\frac{-\sqrt{F \cdot \left(2 \cdot \left(C + \mathsf{hypot}\left(B_m, C\right)\right)\right)}}{B_m}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F -5e-311)
   (/ (sqrt (- F)) (sqrt A))
   (if (<= F 4.2e+25)
     (/ (- (sqrt (* F (* 2.0 (+ C (hypot B_m C)))))) B_m)
     (* (- (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 <= -5e-311) {
		tmp = sqrt(-F) / sqrt(A);
	} else if (F <= 4.2e+25) {
		tmp = -sqrt((F * (2.0 * (C + hypot(B_m, C))))) / B_m;
	} else {
		tmp = -sqrt(2.0) * sqrt((F / 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 <= -5e-311) {
		tmp = Math.sqrt(-F) / Math.sqrt(A);
	} else if (F <= 4.2e+25) {
		tmp = -Math.sqrt((F * (2.0 * (C + Math.hypot(B_m, C))))) / B_m;
	} 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 <= -5e-311:
		tmp = math.sqrt(-F) / math.sqrt(A)
	elif F <= 4.2e+25:
		tmp = -math.sqrt((F * (2.0 * (C + math.hypot(B_m, C))))) / B_m
	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 <= -5e-311)
		tmp = Float64(sqrt(Float64(-F)) / sqrt(A));
	elseif (F <= 4.2e+25)
		tmp = Float64(Float64(-sqrt(Float64(F * Float64(2.0 * Float64(C + hypot(B_m, C)))))) / B_m);
	else
		tmp = Float64(Float64(-sqrt(2.0)) * 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 <= -5e-311)
		tmp = sqrt(-F) / sqrt(A);
	elseif (F <= 4.2e+25)
		tmp = -sqrt((F * (2.0 * (C + hypot(B_m, C))))) / B_m;
	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, -5e-311], N[(N[Sqrt[(-F)], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.2e+25], N[((-N[Sqrt[N[(F * N[(2.0 * N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / B$95$m), $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 < -5.00000000000023e-311

   1. Initial program 37.7%

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

   3. Simplified 45.8%

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

      1. add-sqr-sqrt 45.7%

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

      2. sqrt-unprod 38.1%

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

      3. frac-times 33.8%

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

   5. Applied egg-rr 33.8%

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

   6. Taylor expanded in A around -inf 45.1%

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

      1. mul-1-neg 45.1%

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

   8. Simplified 45.1%

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

      1. distribute-neg-frac 45.1%

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

      2. sqrt-div 60.6%

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

   10. Applied egg-rr 60.6%

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

   if -5.00000000000023e-311 < F < 4.1999999999999998e25

   1. Initial program 18.5%

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

   2. Taylor expanded in A around 0 8.9%

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

      1. mul-1-neg 8.9%

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

      2. *-commutative 8.9%

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

      3. distribute-rgt-neg-in 8.9%

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

      4. unpow2 8.9%

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

      5. unpow2 8.9%

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

      6. hypot-def 20.1%

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

   4. Simplified 20.1%

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

      1. pow1/2 20.1%

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

      2. *-commutative 20.1%

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

      3. unpow-prod-down 20.1%

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

      4. pow1/2 20.1%

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

      5. pow1/2 20.1%

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

   6. Applied egg-rr 20.1%

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

      1. *-commutative 20.1%

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

      2. sqrt-prod 20.1%

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

      3. distribute-rgt-neg-out 20.1%

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

      4. *-commutative 20.1%

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

      5. neg-sub0 20.1%

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

      6. associate-*l/ 20.1%

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

      7. sqrt-unprod 20.2%

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

      8. *-commutative 20.2%

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

   8. Applied egg-rr 20.2%

      \[\leadsto \color{blue}{0 - \frac{\sqrt{2 \cdot \left(\left(C + \mathsf{hypot}\left(B, C\right)\right) \cdot F\right)}}{B}} \]
   9. Step-by-step derivation

      1. neg-sub0 20.2%

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

      2. distribute-frac-neg 20.2%

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

      3. *-commutative 20.2%

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

      4. *-commutative 20.2%

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

      5. associate-*l* 20.2%

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

   10. Simplified 20.2%

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

   if 4.1999999999999998e25 < F

   1. Initial program 14.5%

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

   2. Taylor expanded in C around 0 8.5%

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

      1. mul-1-neg 8.5%

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

      2. *-commutative 8.5%

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

      3. distribute-rgt-neg-in 8.5%

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

      4. +-commutative 8.5%

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

      5. unpow2 8.5%

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

      6. unpow2 8.5%

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

      7. hypot-def 9.8%

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

   4. Simplified 9.8%

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

   5. Taylor expanded in A around 0 19.2%

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

      1. mul-1-neg 19.2%

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

   7. Simplified 19.2%

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

4. Final simplification 23.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{\sqrt{-F}}{\sqrt{A}}\\ \mathbf{elif}\;F \leq 4.2 \cdot 10^{+25}:\\ \;\;\;\;\frac{-\sqrt{F \cdot \left(2 \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)\right)}}{B}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}\\ \end{array} \]

Alternative 13: 40.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := -\sqrt{2}\\ \mathbf{if}\;F \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{\sqrt{-F}}{\sqrt{A}}\\ \mathbf{elif}\;F \leq 10^{-8}:\\ \;\;\;\;\frac{t_0}{B_m} \cdot \sqrt{B_m \cdot F}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \sqrt{\frac{F}{B_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (sqrt 2.0))))
   (if (<= F -5e-311)
     (/ (sqrt (- F)) (sqrt A))
     (if (<= F 1e-8)
       (* (/ t_0 B_m) (sqrt (* B_m F)))
       (* t_0 (sqrt (/ F B_m)))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = -sqrt(2.0);
	double tmp;
	if (F <= -5e-311) {
		tmp = sqrt(-F) / sqrt(A);
	} else if (F <= 1e-8) {
		tmp = (t_0 / B_m) * sqrt((B_m * F));
	} else {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = -sqrt(2.0d0)
    if (f <= (-5d-311)) then
        tmp = sqrt(-f) / sqrt(a)
    else if (f <= 1d-8) then
        tmp = (t_0 / b_m) * sqrt((b_m * f))
    else
        tmp = t_0 * 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 t_0 = -Math.sqrt(2.0);
	double tmp;
	if (F <= -5e-311) {
		tmp = Math.sqrt(-F) / Math.sqrt(A);
	} else if (F <= 1e-8) {
		tmp = (t_0 / B_m) * Math.sqrt((B_m * F));
	} else {
		tmp = t_0 * Math.sqrt((F / B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = -math.sqrt(2.0)
	tmp = 0
	if F <= -5e-311:
		tmp = math.sqrt(-F) / math.sqrt(A)
	elif F <= 1e-8:
		tmp = (t_0 / B_m) * math.sqrt((B_m * F))
	else:
		tmp = t_0 * math.sqrt((F / B_m))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(-sqrt(2.0))
	tmp = 0.0
	if (F <= -5e-311)
		tmp = Float64(sqrt(Float64(-F)) / sqrt(A));
	elseif (F <= 1e-8)
		tmp = Float64(Float64(t_0 / B_m) * sqrt(Float64(B_m * F)));
	else
		tmp = Float64(t_0 * sqrt(Float64(F / 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);
	tmp = 0.0;
	if (F <= -5e-311)
		tmp = sqrt(-F) / sqrt(A);
	elseif (F <= 1e-8)
		tmp = (t_0 / B_m) * sqrt((B_m * F));
	else
		tmp = t_0 * sqrt((F / 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[2.0], $MachinePrecision])}, If[LessEqual[F, -5e-311], N[(N[Sqrt[(-F)], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1e-8], N[(N[(t$95$0 / B$95$m), $MachinePrecision] * N[Sqrt[N[(B$95$m * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.00000000000023e-311

   1. Initial program 37.7%

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

   3. Simplified 45.8%

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

      1. add-sqr-sqrt 45.7%

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

      2. sqrt-unprod 38.1%

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

      3. frac-times 33.8%

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

   5. Applied egg-rr 33.8%

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

   6. Taylor expanded in A around -inf 45.1%

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

      1. mul-1-neg 45.1%

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

   8. Simplified 45.1%

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

      1. distribute-neg-frac 45.1%

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

      2. sqrt-div 60.6%

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

   10. Applied egg-rr 60.6%

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

   if -5.00000000000023e-311 < F < 1e-8

   1. Initial program 19.8%

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

   2. Taylor expanded in C around 0 9.6%

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

      1. mul-1-neg 9.6%

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

      2. *-commutative 9.6%

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

      3. distribute-rgt-neg-in 9.6%

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

      4. +-commutative 9.6%

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

      5. unpow2 9.6%

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

      6. unpow2 9.6%

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

      7. hypot-def 21.6%

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

   4. Simplified 21.6%

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

   5. Taylor expanded in A around 0 18.5%

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

   if 1e-8 < 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 C around 0 7.8%

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

      1. mul-1-neg 7.8%

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

      2. *-commutative 7.8%

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

      3. distribute-rgt-neg-in 7.8%

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

      4. +-commutative 7.8%

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

      5. unpow2 7.8%

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

      6. unpow2 7.8%

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

      7. hypot-def 9.9%

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

   4. Simplified 9.9%

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

   5. Taylor expanded in A around 0 17.8%

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

      1. mul-1-neg 17.8%

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

   7. Simplified 17.8%

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

4. Final simplification 22.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{\sqrt{-F}}{\sqrt{A}}\\ \mathbf{elif}\;F \leq 10^{-8}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{B \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}\\ \end{array} \]

Alternative 14: 15.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \sqrt{-\frac{F}{C}}\\ \mathbf{if}\;C \leq -9.5 \cdot 10^{+44}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;C \leq 5.5 \cdot 10^{-85}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;C \leq 5 \cdot 10^{+25}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-F}}{\sqrt{A}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (sqrt (- (/ F C)))))
   (if (<= C -9.5e+44)
     t_0
     (if (<= C 5.5e-85)
       (sqrt (/ (- F) A))
       (if (<= C 5e+25) t_0 (/ (sqrt (- F)) (sqrt A)))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = sqrt(-(F / C));
	double tmp;
	if (C <= -9.5e+44) {
		tmp = t_0;
	} else if (C <= 5.5e-85) {
		tmp = sqrt((-F / A));
	} else if (C <= 5e+25) {
		tmp = t_0;
	} else {
		tmp = sqrt(-F) / sqrt(A);
	}
	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 / c))
    if (c <= (-9.5d+44)) then
        tmp = t_0
    else if (c <= 5.5d-85) then
        tmp = sqrt((-f / a))
    else if (c <= 5d+25) then
        tmp = t_0
    else
        tmp = sqrt(-f) / sqrt(a)
    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 / C));
	double tmp;
	if (C <= -9.5e+44) {
		tmp = t_0;
	} else if (C <= 5.5e-85) {
		tmp = Math.sqrt((-F / A));
	} else if (C <= 5e+25) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt(-F) / Math.sqrt(A);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = math.sqrt(-(F / C))
	tmp = 0
	if C <= -9.5e+44:
		tmp = t_0
	elif C <= 5.5e-85:
		tmp = math.sqrt((-F / A))
	elif C <= 5e+25:
		tmp = t_0
	else:
		tmp = math.sqrt(-F) / math.sqrt(A)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = sqrt(Float64(-Float64(F / C)))
	tmp = 0.0
	if (C <= -9.5e+44)
		tmp = t_0;
	elseif (C <= 5.5e-85)
		tmp = sqrt(Float64(Float64(-F) / A));
	elseif (C <= 5e+25)
		tmp = t_0;
	else
		tmp = Float64(sqrt(Float64(-F)) / sqrt(A));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = sqrt(-(F / C));
	tmp = 0.0;
	if (C <= -9.5e+44)
		tmp = t_0;
	elseif (C <= 5.5e-85)
		tmp = sqrt((-F / A));
	elseif (C <= 5e+25)
		tmp = t_0;
	else
		tmp = sqrt(-F) / sqrt(A);
	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 / C), $MachinePrecision])], $MachinePrecision]}, If[LessEqual[C, -9.5e+44], t$95$0, If[LessEqual[C, 5.5e-85], N[Sqrt[N[((-F) / A), $MachinePrecision]], $MachinePrecision], If[LessEqual[C, 5e+25], t$95$0, N[(N[Sqrt[(-F)], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if C < -9.5000000000000004e44 or 5.4999999999999997e-85 < C < 5.00000000000000024e25

   1. Initial program 10.5%

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

   3. Simplified 15.6%

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

      1. add-sqr-sqrt 9.2%

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

      2. sqrt-unprod 8.0%

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

      3. frac-times 6.7%

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

   5. Applied egg-rr 6.7%

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

   6. Taylor expanded in A around inf 24.5%

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

      1. mul-1-neg 24.5%

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

   8. Simplified 24.5%

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

   if -9.5000000000000004e44 < C < 5.4999999999999997e-85

   1. Initial program 24.4%

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

   3. Simplified 29.0%

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

      1. add-sqr-sqrt 2.5%

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

      2. sqrt-unprod 3.0%

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

      3. frac-times 2.8%

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

   5. Applied egg-rr 2.8%

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

   6. Taylor expanded in A around -inf 15.2%

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

      1. mul-1-neg 15.2%

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

   8. Simplified 15.2%

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

   if 5.00000000000000024e25 < C

   1. Initial program 19.5%

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

   3. Simplified 28.3%

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

      1. add-sqr-sqrt 7.2%

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

      2. sqrt-unprod 5.6%

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

      3. frac-times 5.5%

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

   5. Applied egg-rr 5.5%

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

   6. Taylor expanded in A around -inf 17.5%

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

      1. mul-1-neg 17.5%

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

   8. Simplified 17.5%

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

      1. distribute-neg-frac 17.5%

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

      2. sqrt-div 22.7%

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

   10. Applied egg-rr 22.7%

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

4. Final simplification 19.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -9.5 \cdot 10^{+44}:\\ \;\;\;\;\sqrt{-\frac{F}{C}}\\ \mathbf{elif}\;C \leq 5.5 \cdot 10^{-85}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;C \leq 5 \cdot 10^{+25}:\\ \;\;\;\;\sqrt{-\frac{F}{C}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-F}}{\sqrt{A}}\\ \end{array} \]

Alternative 15: 33.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{\sqrt{-F}}{\sqrt{A}}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F -5e-311)
   (/ (sqrt (- F)) (sqrt A))
   (* (- (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 <= -5e-311) {
		tmp = sqrt(-F) / sqrt(A);
	} 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 <= (-5d-311)) then
        tmp = sqrt(-f) / sqrt(a)
    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 <= -5e-311) {
		tmp = Math.sqrt(-F) / Math.sqrt(A);
	} 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 <= -5e-311:
		tmp = math.sqrt(-F) / math.sqrt(A)
	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 <= -5e-311)
		tmp = Float64(sqrt(Float64(-F)) / sqrt(A));
	else
		tmp = Float64(Float64(-sqrt(2.0)) * 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 <= -5e-311)
		tmp = sqrt(-F) / sqrt(A);
	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, -5e-311], N[(N[Sqrt[(-F)], $MachinePrecision] / N[Sqrt[A], $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 < -5.00000000000023e-311

   1. Initial program 37.7%

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

   3. Simplified 45.8%

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

      1. add-sqr-sqrt 45.7%

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

      2. sqrt-unprod 38.1%

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

      3. frac-times 33.8%

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

   5. Applied egg-rr 33.8%

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

   6. Taylor expanded in A around -inf 45.1%

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

      1. mul-1-neg 45.1%

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

   8. Simplified 45.1%

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

      1. distribute-neg-frac 45.1%

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

      2. sqrt-div 60.6%

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

   10. Applied egg-rr 60.6%

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

   if -5.00000000000023e-311 < F

   1. Initial program 16.7%

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

   2. Taylor expanded in C around 0 8.7%

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

      1. mul-1-neg 8.7%

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

      2. *-commutative 8.7%

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

      3. distribute-rgt-neg-in 8.7%

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

      4. +-commutative 8.7%

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

      5. unpow2 8.7%

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

      6. unpow2 8.7%

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

      7. hypot-def 15.4%

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

   4. Simplified 15.4%

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

   5. Taylor expanded in A around 0 14.7%

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

      1. mul-1-neg 14.7%

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

   7. Simplified 14.7%

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

4. Final simplification 19.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{\sqrt{-F}}{\sqrt{A}}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}\\ \end{array} \]

Alternative 16: 14.6% accurate, 5.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;C \leq -3.1 \cdot 10^{+45} \lor \neg \left(C \leq 2.9 \cdot 10^{-83}\right) \land C \leq 2.15 \cdot 10^{+26}:\\ \;\;\;\;\sqrt{-\frac{F}{C}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (or (<= C -3.1e+45) (and (not (<= C 2.9e-83)) (<= C 2.15e+26)))
   (sqrt (- (/ F C)))
   (sqrt (/ (- F) A))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if ((C <= -3.1e+45) || (!(C <= 2.9e-83) && (C <= 2.15e+26))) {
		tmp = sqrt(-(F / C));
	} else {
		tmp = sqrt((-F / A));
	}
	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 ((c <= (-3.1d+45)) .or. (.not. (c <= 2.9d-83)) .and. (c <= 2.15d+26)) then
        tmp = sqrt(-(f / c))
    else
        tmp = sqrt((-f / a))
    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 ((C <= -3.1e+45) || (!(C <= 2.9e-83) && (C <= 2.15e+26))) {
		tmp = Math.sqrt(-(F / C));
	} else {
		tmp = Math.sqrt((-F / A));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if (C <= -3.1e+45) or (not (C <= 2.9e-83) and (C <= 2.15e+26)):
		tmp = math.sqrt(-(F / C))
	else:
		tmp = math.sqrt((-F / A))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if ((C <= -3.1e+45) || (!(C <= 2.9e-83) && (C <= 2.15e+26)))
		tmp = sqrt(Float64(-Float64(F / C)));
	else
		tmp = sqrt(Float64(Float64(-F) / A));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if ((C <= -3.1e+45) || (~((C <= 2.9e-83)) && (C <= 2.15e+26)))
		tmp = sqrt(-(F / C));
	else
		tmp = sqrt((-F / A));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[Or[LessEqual[C, -3.1e+45], And[N[Not[LessEqual[C, 2.9e-83]], $MachinePrecision], LessEqual[C, 2.15e+26]]], N[Sqrt[(-N[(F / C), $MachinePrecision])], $MachinePrecision], N[Sqrt[N[((-F) / A), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < -3.09999999999999988e45 or 2.8999999999999999e-83 < C < 2.1499999999999999e26

   1. Initial program 10.5%

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

   3. Simplified 15.6%

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

      1. add-sqr-sqrt 9.2%

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

      2. sqrt-unprod 8.0%

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

      3. frac-times 6.7%

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

   5. Applied egg-rr 6.7%

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

   6. Taylor expanded in A around inf 24.5%

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

      1. mul-1-neg 24.5%

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

   8. Simplified 24.5%

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

   if -3.09999999999999988e45 < C < 2.8999999999999999e-83 or 2.1499999999999999e26 < C

   1. Initial program 23.1%

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

   3. Simplified 28.8%

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

      1. add-sqr-sqrt 3.7%

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

      2. sqrt-unprod 3.7%

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

      3. frac-times 3.5%

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

   5. Applied egg-rr 3.5%

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

   6. Taylor expanded in A around -inf 15.8%

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

      1. mul-1-neg 15.8%

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

   8. Simplified 15.8%

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

4. Final simplification 18.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -3.1 \cdot 10^{+45} \lor \neg \left(C \leq 2.9 \cdot 10^{-83}\right) \land C \leq 2.15 \cdot 10^{+26}:\\ \;\;\;\;\sqrt{-\frac{F}{C}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \end{array} \]

Alternative 17: 11.3% 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 18.6%

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

3. Simplified 24.1%

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

   1. add-sqr-sqrt 5.7%

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

   2. sqrt-unprod 5.2%

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

   3. frac-times 4.6%

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

5. Applied egg-rr 4.6%

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

6. Taylor expanded in A around -inf 12.7%

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

   1. mul-1-neg 12.7%

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

8. Simplified 12.7%

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

9. Final simplification 12.7%

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

Alternative 18: 1.2% accurate, 6.2× 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(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 18.6%

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

3. Simplified 24.1%

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

   1. add-sqr-sqrt 5.7%

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

   2. sqrt-unprod 5.2%

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

   3. frac-times 4.6%

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

5. Applied egg-rr 4.6%

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

6. Taylor expanded in A around -inf 12.7%

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

   1. mul-1-neg 12.7%

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

8. Simplified 12.7%

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

   1. expm1-log1p-u 12.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{-\frac{F}{A}}\right)\right)} \]

   2. expm1-udef 6.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{-\frac{F}{A}}\right)} - 1} \]

   3. add-sqr-sqrt 6.4%

      \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\sqrt{-\frac{F}{A}} \cdot \sqrt{-\frac{F}{A}}}}\right)} - 1 \]

   4. sqrt-unprod 6.7%

      \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\sqrt{\left(-\frac{F}{A}\right) \cdot \left(-\frac{F}{A}\right)}}}\right)} - 1 \]

   5. sqr-neg 6.7%

      \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\sqrt{\color{blue}{\frac{F}{A} \cdot \frac{F}{A}}}}\right)} - 1 \]

   6. sqrt-unprod 1.3%

      \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\sqrt{\frac{F}{A}} \cdot \sqrt{\frac{F}{A}}}}\right)} - 1 \]

   7. add-sqr-sqrt 1.3%

      \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{F}{A}}}\right)} - 1 \]

10. Applied egg-rr 1.3%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{F}{A}}\right)} - 1} \]
11. Step-by-step derivation

    1. expm1-def 1.1%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{F}{A}}\right)\right)} \]

    2. expm1-log1p 1.1%

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

12. Simplified 1.1%

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

13. Final simplification 1.1%

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

Reproduce

herbie shell --seed 2023318 
(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))))