ABCF->ab-angle a

Percentage Accurate: 20.5% → 54.4%

Time: 18.7s

Alternatives: 12

Speedup: 6.0×

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: 20.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ \frac{-\sqrt{\left(2 \cdot \left(t\_0 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{t\_0} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 54.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{-294}:\\ \;\;\;\;-\sqrt{\left|-0.5 \cdot \left(2 \cdot \frac{F}{A}\right)\right|}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{-230}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot t\_0\right) \cdot \left(C \cdot 4\right)}}{-t\_0}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+233}:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(A + C\right) + \mathsf{hypot}\left(B\_m, A - C\right)}{\mathsf{fma}\left(-4, A \cdot C, {B\_m}^{2}\right)}} \cdot \left(-e^{\log 2 \cdot 0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\_m\right)} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= (pow B_m 2.0) 1e-294)
     (- (sqrt (fabs (* -0.5 (* 2.0 (/ F A))))))
     (if (<= (pow B_m 2.0) 5e-230)
       (/ (sqrt (* (* F t_0) (* C 4.0))) (- t_0))
       (if (<= (pow B_m 2.0) 5e+233)
         (*
          (sqrt
           (*
            F
            (/
             (+ (+ A C) (hypot B_m (- A C)))
             (fma -4.0 (* A C) (pow B_m 2.0)))))
          (- (exp (* (log 2.0) 0.5))))
         (*
          (* (sqrt (+ C (hypot C B_m))) (sqrt F))
          (/ (sqrt 2.0) (- B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (pow(B_m, 2.0) <= 1e-294) {
		tmp = -sqrt(fabs((-0.5 * (2.0 * (F / A)))));
	} else if (pow(B_m, 2.0) <= 5e-230) {
		tmp = sqrt(((F * t_0) * (C * 4.0))) / -t_0;
	} else if (pow(B_m, 2.0) <= 5e+233) {
		tmp = sqrt((F * (((A + C) + hypot(B_m, (A - C))) / fma(-4.0, (A * C), pow(B_m, 2.0))))) * -exp((log(2.0) * 0.5));
	} else {
		tmp = (sqrt((C + hypot(C, B_m))) * sqrt(F)) * (sqrt(2.0) / -B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-294)
		tmp = Float64(-sqrt(abs(Float64(-0.5 * Float64(2.0 * Float64(F / A))))));
	elseif ((B_m ^ 2.0) <= 5e-230)
		tmp = Float64(sqrt(Float64(Float64(F * t_0) * Float64(C * 4.0))) / Float64(-t_0));
	elseif ((B_m ^ 2.0) <= 5e+233)
		tmp = Float64(sqrt(Float64(F * Float64(Float64(Float64(A + C) + hypot(B_m, Float64(A - C))) / fma(-4.0, Float64(A * C), (B_m ^ 2.0))))) * Float64(-exp(Float64(log(2.0) * 0.5))));
	else
		tmp = Float64(Float64(sqrt(Float64(C + hypot(C, B_m))) * sqrt(F)) * Float64(sqrt(2.0) / Float64(-B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-294], (-N[Sqrt[N[Abs[N[(-0.5 * N[(2.0 * N[(F / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-230], N[(N[Sqrt[N[(N[(F * t$95$0), $MachinePrecision] * N[(C * 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+233], N[(N[Sqrt[N[(F * N[(N[(N[(A + C), $MachinePrecision] + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] / N[(-4.0 * N[(A * C), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Exp[N[(N[Log[2.0], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[(N[Sqrt[N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 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 #s(literal 2 binary64)) < 1.00000000000000002e-294

   1. Initial program 20.5%

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

   3. Taylor expanded in F around 0 22.1%

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

   5. Simplified 34.5%

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

   6. Taylor expanded in C around inf 29.8%

      \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{-0.5 \cdot \frac{F}{A}}} \]
   7. Step-by-step derivation

      1. *-commutative 29.8%

         \[\leadsto -\color{blue}{\sqrt{-0.5 \cdot \frac{F}{A}} \cdot \sqrt{2}} \]

      2. pow1/2 29.9%

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

      3. pow1/2 29.9%

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

      4. pow-prod-down 30.0%

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

      5. associate-*r/ 30.0%

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

   8. Applied egg-rr 30.0%

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

      1. unpow1/2 29.9%

         \[\leadsto -\color{blue}{\sqrt{\frac{-0.5 \cdot F}{A} \cdot 2}} \]

   10. Simplified 29.9%

       \[\leadsto -\color{blue}{\sqrt{\frac{-0.5 \cdot F}{A} \cdot 2}} \]
   11. Step-by-step derivation

       1. add-sqr-sqrt 29.9%

          \[\leadsto -\sqrt{\color{blue}{\sqrt{\frac{-0.5 \cdot F}{A} \cdot 2} \cdot \sqrt{\frac{-0.5 \cdot F}{A} \cdot 2}}} \]

       2. pow1/2 29.9%

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

       3. pow1/2 30.0%

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

       4. pow-prod-down 25.6%

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

       5. pow2 25.6%

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

       6. associate-/l* 25.6%

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

   12. Applied egg-rr 25.6%

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

       1. unpow1/2 25.6%

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

       2. unpow2 25.6%

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

       3. rem-sqrt-square 31.8%

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

       4. associate-*l* 31.8%

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

   14. Simplified 31.8%

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

   if 1.00000000000000002e-294 < (pow.f64 B #s(literal 2 binary64)) < 5.00000000000000035e-230

   1. Initial program 20.9%

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

   3. Simplified 41.6%

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

   5. Taylor expanded in A around -inf 32.8%

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

      1. *-commutative 32.8%

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

   7. Simplified 32.8%

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

   if 5.00000000000000035e-230 < (pow.f64 B #s(literal 2 binary64)) < 5.00000000000000009e233

   1. Initial program 31.3%

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

   3. Taylor expanded in F around 0 28.7%

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

   5. Simplified 50.5%

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

      1. pow1/2 50.5%

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

      2. pow-to-exp 50.6%

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

   7. Applied egg-rr 50.6%

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

   if 5.00000000000000009e233 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 5.1%

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

   3. Taylor expanded in A around 0 6.7%

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

      1. mul-1-neg 6.7%

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

      2. *-commutative 6.7%

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

      3. *-commutative 6.7%

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

      4. +-commutative 6.7%

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

      5. unpow2 6.7%

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

      6. unpow2 6.7%

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

      7. hypot-define 27.2%

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

   5. Simplified 27.2%

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

      1. sqrt-prod 39.4%

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

   7. Applied egg-rr 39.4%

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

4. Final simplification 40.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-294}:\\ \;\;\;\;-\sqrt{\left|-0.5 \cdot \left(2 \cdot \frac{F}{A}\right)\right|}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{-230}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(C \cdot 4\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+233}:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}} \cdot \left(-e^{\log 2 \cdot 0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 54.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{-294}:\\ \;\;\;\;-\sqrt{\left|-0.5 \cdot \left(2 \cdot \frac{F}{A}\right)\right|}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{-230}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot t\_0\right) \cdot \left(C \cdot 4\right)}}{-t\_0}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+233}:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(A + C\right) + \mathsf{hypot}\left(B\_m, A - C\right)}{\mathsf{fma}\left(-4, A \cdot C, B\_m \cdot B\_m\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\_m\right)} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= (pow B_m 2.0) 1e-294)
     (- (sqrt (fabs (* -0.5 (* 2.0 (/ F A))))))
     (if (<= (pow B_m 2.0) 5e-230)
       (/ (sqrt (* (* F t_0) (* C 4.0))) (- t_0))
       (if (<= (pow B_m 2.0) 5e+233)
         (*
          (sqrt
           (*
            F
            (/
             (+ (+ A C) (hypot B_m (- A C)))
             (fma -4.0 (* A C) (* B_m B_m)))))
          (- (sqrt 2.0)))
         (*
          (* (sqrt (+ C (hypot C B_m))) (sqrt F))
          (/ (sqrt 2.0) (- B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (pow(B_m, 2.0) <= 1e-294) {
		tmp = -sqrt(fabs((-0.5 * (2.0 * (F / A)))));
	} else if (pow(B_m, 2.0) <= 5e-230) {
		tmp = sqrt(((F * t_0) * (C * 4.0))) / -t_0;
	} else if (pow(B_m, 2.0) <= 5e+233) {
		tmp = sqrt((F * (((A + C) + hypot(B_m, (A - C))) / fma(-4.0, (A * C), (B_m * B_m))))) * -sqrt(2.0);
	} else {
		tmp = (sqrt((C + hypot(C, B_m))) * sqrt(F)) * (sqrt(2.0) / -B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-294)
		tmp = Float64(-sqrt(abs(Float64(-0.5 * Float64(2.0 * Float64(F / A))))));
	elseif ((B_m ^ 2.0) <= 5e-230)
		tmp = Float64(sqrt(Float64(Float64(F * t_0) * Float64(C * 4.0))) / Float64(-t_0));
	elseif ((B_m ^ 2.0) <= 5e+233)
		tmp = Float64(sqrt(Float64(F * Float64(Float64(Float64(A + C) + hypot(B_m, Float64(A - C))) / fma(-4.0, Float64(A * C), Float64(B_m * B_m))))) * Float64(-sqrt(2.0)));
	else
		tmp = Float64(Float64(sqrt(Float64(C + hypot(C, B_m))) * sqrt(F)) * Float64(sqrt(2.0) / Float64(-B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-294], (-N[Sqrt[N[Abs[N[(-0.5 * N[(2.0 * N[(F / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-230], N[(N[Sqrt[N[(N[(F * t$95$0), $MachinePrecision] * N[(C * 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+233], N[(N[Sqrt[N[(F * N[(N[(N[(A + C), $MachinePrecision] + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] / N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], N[(N[(N[Sqrt[N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 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 #s(literal 2 binary64)) < 1.00000000000000002e-294

   1. Initial program 20.5%

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

   3. Taylor expanded in F around 0 22.1%

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

   5. Simplified 34.5%

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

   6. Taylor expanded in C around inf 29.8%

      \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{-0.5 \cdot \frac{F}{A}}} \]
   7. Step-by-step derivation

      1. *-commutative 29.8%

         \[\leadsto -\color{blue}{\sqrt{-0.5 \cdot \frac{F}{A}} \cdot \sqrt{2}} \]

      2. pow1/2 29.9%

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

      3. pow1/2 29.9%

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

      4. pow-prod-down 30.0%

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

      5. associate-*r/ 30.0%

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

   8. Applied egg-rr 30.0%

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

      1. unpow1/2 29.9%

         \[\leadsto -\color{blue}{\sqrt{\frac{-0.5 \cdot F}{A} \cdot 2}} \]

   10. Simplified 29.9%

       \[\leadsto -\color{blue}{\sqrt{\frac{-0.5 \cdot F}{A} \cdot 2}} \]
   11. Step-by-step derivation

       1. add-sqr-sqrt 29.9%

          \[\leadsto -\sqrt{\color{blue}{\sqrt{\frac{-0.5 \cdot F}{A} \cdot 2} \cdot \sqrt{\frac{-0.5 \cdot F}{A} \cdot 2}}} \]

       2. pow1/2 29.9%

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

       3. pow1/2 30.0%

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

       4. pow-prod-down 25.6%

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

       5. pow2 25.6%

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

       6. associate-/l* 25.6%

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

   12. Applied egg-rr 25.6%

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

       1. unpow1/2 25.6%

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

       2. unpow2 25.6%

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

       3. rem-sqrt-square 31.8%

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

       4. associate-*l* 31.8%

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

   14. Simplified 31.8%

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

   if 1.00000000000000002e-294 < (pow.f64 B #s(literal 2 binary64)) < 5.00000000000000035e-230

   1. Initial program 20.9%

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

   3. Simplified 41.6%

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

   5. Taylor expanded in A around -inf 32.8%

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

      1. *-commutative 32.8%

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

   7. Simplified 32.8%

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

   if 5.00000000000000035e-230 < (pow.f64 B #s(literal 2 binary64)) < 5.00000000000000009e233

   1. Initial program 31.3%

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

   3. Taylor expanded in F around 0 28.7%

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

   5. Simplified 50.5%

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

      1. unpow2 50.5%

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

   7. Applied egg-rr 50.5%

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

   if 5.00000000000000009e233 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 5.1%

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

   3. Taylor expanded in A around 0 6.7%

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

      1. mul-1-neg 6.7%

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

      2. *-commutative 6.7%

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

      3. *-commutative 6.7%

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

      4. +-commutative 6.7%

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

      5. unpow2 6.7%

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

      6. unpow2 6.7%

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

      7. hypot-define 27.2%

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

   5. Simplified 27.2%

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

      1. sqrt-prod 39.4%

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

   7. Applied egg-rr 39.4%

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

4. Final simplification 40.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-294}:\\ \;\;\;\;-\sqrt{\left|-0.5 \cdot \left(2 \cdot \frac{F}{A}\right)\right|}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{-230}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(C \cdot 4\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+233}:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 52.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 10^{-155}:\\ \;\;\;\;-\sqrt{\left|-0.5 \cdot \left(2 \cdot \frac{F}{A}\right)\right|}\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{+100}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{F \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B\_m, A - C\right)\right)}{\mathsf{fma}\left(-4, A \cdot C, {B\_m}^{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\_m\right)} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 1e-155)
   (- (sqrt (fabs (* -0.5 (* 2.0 (/ F A))))))
   (if (<= (pow B_m 2.0) 1e+100)
     (-
      (sqrt
       (*
        2.0
        (/
         (* F (+ (+ A C) (hypot B_m (- A C))))
         (fma -4.0 (* A C) (pow B_m 2.0))))))
     (* (* (sqrt (+ C (hypot C B_m))) (sqrt F)) (/ (sqrt 2.0) (- B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 1e-155) {
		tmp = -sqrt(fabs((-0.5 * (2.0 * (F / A)))));
	} else if (pow(B_m, 2.0) <= 1e+100) {
		tmp = -sqrt((2.0 * ((F * ((A + C) + hypot(B_m, (A - C)))) / fma(-4.0, (A * C), pow(B_m, 2.0)))));
	} else {
		tmp = (sqrt((C + hypot(C, B_m))) * sqrt(F)) * (sqrt(2.0) / -B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-155)
		tmp = Float64(-sqrt(abs(Float64(-0.5 * Float64(2.0 * Float64(F / A))))));
	elseif ((B_m ^ 2.0) <= 1e+100)
		tmp = Float64(-sqrt(Float64(2.0 * Float64(Float64(F * Float64(Float64(A + C) + hypot(B_m, Float64(A - C)))) / fma(-4.0, Float64(A * C), (B_m ^ 2.0))))));
	else
		tmp = Float64(Float64(sqrt(Float64(C + hypot(C, B_m))) * sqrt(F)) * Float64(sqrt(2.0) / Float64(-B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-155], (-N[Sqrt[N[Abs[N[(-0.5 * N[(2.0 * N[(F / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+100], (-N[Sqrt[N[(2.0 * N[(N[(F * N[(N[(A + C), $MachinePrecision] + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-4.0 * N[(A * C), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), N[(N[(N[Sqrt[N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 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 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.00000000000000001e-155

   1. Initial program 22.7%

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

   3. Taylor expanded in F around 0 21.7%

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

   5. Simplified 35.6%

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

   6. Taylor expanded in C around inf 27.6%

      \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{-0.5 \cdot \frac{F}{A}}} \]
   7. Step-by-step derivation

      1. *-commutative 27.6%

         \[\leadsto -\color{blue}{\sqrt{-0.5 \cdot \frac{F}{A}} \cdot \sqrt{2}} \]

      2. pow1/2 27.8%

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

      3. pow1/2 27.8%

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

      4. pow-prod-down 27.9%

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

      5. associate-*r/ 27.9%

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

   8. Applied egg-rr 27.9%

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

      1. unpow1/2 27.7%

         \[\leadsto -\color{blue}{\sqrt{\frac{-0.5 \cdot F}{A} \cdot 2}} \]

   10. Simplified 27.7%

       \[\leadsto -\color{blue}{\sqrt{\frac{-0.5 \cdot F}{A} \cdot 2}} \]
   11. Step-by-step derivation

       1. add-sqr-sqrt 27.7%

          \[\leadsto -\sqrt{\color{blue}{\sqrt{\frac{-0.5 \cdot F}{A} \cdot 2} \cdot \sqrt{\frac{-0.5 \cdot F}{A} \cdot 2}}} \]

       2. pow1/2 27.7%

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

       3. pow1/2 27.9%

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

       4. pow-prod-down 25.2%

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

       5. pow2 25.2%

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

       6. associate-/l* 25.2%

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

   12. Applied egg-rr 25.2%

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

       1. unpow1/2 25.2%

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

       2. unpow2 25.2%

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

       3. rem-sqrt-square 29.8%

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

       4. associate-*l* 29.8%

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

   14. Simplified 29.8%

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

   if 1.00000000000000001e-155 < (pow.f64 B #s(literal 2 binary64)) < 1.00000000000000002e100

   1. Initial program 33.9%

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

   3. Taylor expanded in F around 0 32.4%

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

   5. Simplified 48.3%

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

      1. *-commutative 48.3%

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

      2. pow1/2 48.3%

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

      3. pow1/2 48.3%

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

      4. pow-prod-down 48.4%

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

      5. associate-*r/ 46.8%

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

      6. +-commutative 46.8%

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

   7. Applied egg-rr 46.8%

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

      1. unpow1/2 46.8%

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

   9. Simplified 46.8%

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

   if 1.00000000000000002e100 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 9.0%

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

   3. Taylor expanded in A around 0 9.1%

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

      1. mul-1-neg 9.1%

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

      2. *-commutative 9.1%

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

      3. *-commutative 9.1%

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

      4. +-commutative 9.1%

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

      5. unpow2 9.1%

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

      6. unpow2 9.1%

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

      7. hypot-define 25.7%

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

   5. Simplified 25.7%

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

      1. sqrt-prod 36.2%

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

   7. Applied egg-rr 36.2%

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

4. Final simplification 35.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-155}:\\ \;\;\;\;-\sqrt{\left|-0.5 \cdot \left(2 \cdot \frac{F}{A}\right)\right|}\\ \mathbf{elif}\;{B}^{2} \leq 10^{+100}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{F \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 50.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 5000000000000:\\ \;\;\;\;-\sqrt{\left|-0.5 \cdot \left(2 \cdot \frac{F}{A}\right)\right|}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+150}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(-\sqrt{F \cdot \left(A + \mathsf{hypot}\left(B\_m, A\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\_m\right)} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 5000000000000.0)
   (- (sqrt (fabs (* -0.5 (* 2.0 (/ F A))))))
   (if (<= (pow B_m 2.0) 5e+150)
     (* (/ (sqrt 2.0) B_m) (- (sqrt (* F (+ A (hypot B_m A))))))
     (* (* (sqrt (+ C (hypot C B_m))) (sqrt F)) (/ (sqrt 2.0) (- B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 5000000000000.0) {
		tmp = -sqrt(fabs((-0.5 * (2.0 * (F / A)))));
	} else if (pow(B_m, 2.0) <= 5e+150) {
		tmp = (sqrt(2.0) / B_m) * -sqrt((F * (A + hypot(B_m, A))));
	} else {
		tmp = (sqrt((C + hypot(C, B_m))) * sqrt(F)) * (sqrt(2.0) / -B_m);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (Math.pow(B_m, 2.0) <= 5000000000000.0) {
		tmp = -Math.sqrt(Math.abs((-0.5 * (2.0 * (F / A)))));
	} else if (Math.pow(B_m, 2.0) <= 5e+150) {
		tmp = (Math.sqrt(2.0) / B_m) * -Math.sqrt((F * (A + Math.hypot(B_m, A))));
	} else {
		tmp = (Math.sqrt((C + Math.hypot(C, B_m))) * Math.sqrt(F)) * (Math.sqrt(2.0) / -B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if math.pow(B_m, 2.0) <= 5000000000000.0:
		tmp = -math.sqrt(math.fabs((-0.5 * (2.0 * (F / A)))))
	elif math.pow(B_m, 2.0) <= 5e+150:
		tmp = (math.sqrt(2.0) / B_m) * -math.sqrt((F * (A + math.hypot(B_m, A))))
	else:
		tmp = (math.sqrt((C + math.hypot(C, B_m))) * math.sqrt(F)) * (math.sqrt(2.0) / -B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5000000000000.0)
		tmp = Float64(-sqrt(abs(Float64(-0.5 * Float64(2.0 * Float64(F / A))))));
	elseif ((B_m ^ 2.0) <= 5e+150)
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(-sqrt(Float64(F * Float64(A + hypot(B_m, A))))));
	else
		tmp = Float64(Float64(sqrt(Float64(C + hypot(C, B_m))) * sqrt(F)) * Float64(sqrt(2.0) / Float64(-B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 5000000000000.0)
		tmp = -sqrt(abs((-0.5 * (2.0 * (F / A)))));
	elseif ((B_m ^ 2.0) <= 5e+150)
		tmp = (sqrt(2.0) / B_m) * -sqrt((F * (A + hypot(B_m, A))));
	else
		tmp = (sqrt((C + hypot(C, B_m))) * sqrt(F)) * (sqrt(2.0) / -B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5000000000000.0], (-N[Sqrt[N[Abs[N[(-0.5 * N[(2.0 * N[(F / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+150], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[N[(F * N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[(N[Sqrt[N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 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 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 5e12

   1. Initial program 25.3%

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

   3. Taylor expanded in F around 0 23.9%

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

   5. Simplified 38.3%

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

   6. Taylor expanded in C around inf 26.3%

      \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{-0.5 \cdot \frac{F}{A}}} \]
   7. Step-by-step derivation

      1. *-commutative 26.3%

         \[\leadsto -\color{blue}{\sqrt{-0.5 \cdot \frac{F}{A}} \cdot \sqrt{2}} \]

      2. pow1/2 26.4%

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

      3. pow1/2 26.4%

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

      4. pow-prod-down 26.5%

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

      5. associate-*r/ 26.5%

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

   8. Applied egg-rr 26.5%

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

      1. unpow1/2 26.4%

         \[\leadsto -\color{blue}{\sqrt{\frac{-0.5 \cdot F}{A} \cdot 2}} \]

   10. Simplified 26.4%

       \[\leadsto -\color{blue}{\sqrt{\frac{-0.5 \cdot F}{A} \cdot 2}} \]
   11. Step-by-step derivation

       1. add-sqr-sqrt 26.4%

          \[\leadsto -\sqrt{\color{blue}{\sqrt{\frac{-0.5 \cdot F}{A} \cdot 2} \cdot \sqrt{\frac{-0.5 \cdot F}{A} \cdot 2}}} \]

       2. pow1/2 26.4%

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

       3. pow1/2 26.5%

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

       4. pow-prod-down 24.3%

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

       5. pow2 24.3%

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

       6. associate-/l* 24.3%

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

   12. Applied egg-rr 24.3%

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

       1. unpow1/2 24.3%

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

       2. unpow2 24.3%

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

       3. rem-sqrt-square 28.4%

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

       4. associate-*l* 28.4%

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

   14. Simplified 28.4%

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

   if 5e12 < (pow.f64 B #s(literal 2 binary64)) < 5.00000000000000009e150

   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. Add Preprocessing

   3. Taylor expanded in C around 0 23.1%

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

      1. mul-1-neg 23.1%

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

      2. *-commutative 23.1%

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

      3. *-commutative 23.1%

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

      4. +-commutative 23.1%

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

      5. unpow2 23.1%

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

      6. unpow2 23.1%

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

      7. hypot-define 30.3%

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

   5. Simplified 30.3%

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

   if 5.00000000000000009e150 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 6.1%

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

   3. Taylor expanded in A around 0 7.4%

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

      1. mul-1-neg 7.4%

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

      2. *-commutative 7.4%

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

      3. *-commutative 7.4%

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

      4. +-commutative 7.4%

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

      5. unpow2 7.4%

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

      6. unpow2 7.4%

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

      7. hypot-define 25.9%

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

   5. Simplified 25.9%

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

      1. sqrt-prod 37.5%

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

   7. Applied egg-rr 37.5%

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

4. Final simplification 31.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5000000000000:\\ \;\;\;\;-\sqrt{\left|-0.5 \cdot \left(2 \cdot \frac{F}{A}\right)\right|}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+150}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 43.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 10^{+261}:\\ \;\;\;\;-\sqrt{\left|-0.5 \cdot \left(2 \cdot \frac{F}{A}\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{\frac{1 + \left(\frac{A}{B\_m} + \frac{C}{B\_m}\right)}{B\_m}}\right) \cdot \left(-\sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 1e+261)
   (- (sqrt (fabs (* -0.5 (* 2.0 (/ F A))))))
   (*
    (* (sqrt F) (sqrt (/ (+ 1.0 (+ (/ A B_m) (/ C B_m))) B_m)))
    (- (sqrt 2.0)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 1e+261) {
		tmp = -sqrt(fabs((-0.5 * (2.0 * (F / A)))));
	} else {
		tmp = (sqrt(F) * sqrt(((1.0 + ((A / B_m) + (C / B_m))) / B_m))) * -sqrt(2.0);
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
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 ((b_m ** 2.0d0) <= 1d+261) then
        tmp = -sqrt(abs(((-0.5d0) * (2.0d0 * (f / a)))))
    else
        tmp = (sqrt(f) * sqrt(((1.0d0 + ((a / b_m) + (c / b_m))) / b_m))) * -sqrt(2.0d0)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (Math.pow(B_m, 2.0) <= 1e+261) {
		tmp = -Math.sqrt(Math.abs((-0.5 * (2.0 * (F / A)))));
	} else {
		tmp = (Math.sqrt(F) * Math.sqrt(((1.0 + ((A / B_m) + (C / B_m))) / B_m))) * -Math.sqrt(2.0);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if math.pow(B_m, 2.0) <= 1e+261:
		tmp = -math.sqrt(math.fabs((-0.5 * (2.0 * (F / A)))))
	else:
		tmp = (math.sqrt(F) * math.sqrt(((1.0 + ((A / B_m) + (C / B_m))) / B_m))) * -math.sqrt(2.0)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e+261)
		tmp = Float64(-sqrt(abs(Float64(-0.5 * Float64(2.0 * Float64(F / A))))));
	else
		tmp = Float64(Float64(sqrt(F) * sqrt(Float64(Float64(1.0 + Float64(Float64(A / B_m) + Float64(C / B_m))) / B_m))) * Float64(-sqrt(2.0)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 1e+261)
		tmp = -sqrt(abs((-0.5 * (2.0 * (F / A)))));
	else
		tmp = (sqrt(F) * sqrt(((1.0 + ((A / B_m) + (C / B_m))) / B_m))) * -sqrt(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+261], (-N[Sqrt[N[Abs[N[(-0.5 * N[(2.0 * N[(F / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(N[(1.0 + N[(N[(A / B$95$m), $MachinePrecision] + N[(C / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.9999999999999993e260

   1. Initial program 25.0%

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

   3. Taylor expanded in F around 0 24.5%

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

   5. Simplified 41.4%

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

   6. Taylor expanded in C around inf 26.2%

      \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{-0.5 \cdot \frac{F}{A}}} \]
   7. Step-by-step derivation

      1. *-commutative 26.2%

         \[\leadsto -\color{blue}{\sqrt{-0.5 \cdot \frac{F}{A}} \cdot \sqrt{2}} \]

      2. pow1/2 26.4%

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

      3. pow1/2 26.4%

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

      4. pow-prod-down 26.5%

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

      5. associate-*r/ 26.5%

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

   8. Applied egg-rr 26.5%

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

      1. unpow1/2 26.3%

         \[\leadsto -\color{blue}{\sqrt{\frac{-0.5 \cdot F}{A} \cdot 2}} \]

   10. Simplified 26.3%

       \[\leadsto -\color{blue}{\sqrt{\frac{-0.5 \cdot F}{A} \cdot 2}} \]
   11. Step-by-step derivation

       1. add-sqr-sqrt 26.3%

          \[\leadsto -\sqrt{\color{blue}{\sqrt{\frac{-0.5 \cdot F}{A} \cdot 2} \cdot \sqrt{\frac{-0.5 \cdot F}{A} \cdot 2}}} \]

       2. pow1/2 26.3%

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

       3. pow1/2 26.5%

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

       4. pow-prod-down 23.7%

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

       5. pow2 23.7%

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

       6. associate-/l* 23.7%

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

   12. Applied egg-rr 23.7%

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

       1. unpow1/2 23.7%

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

       2. unpow2 23.7%

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

       3. rem-sqrt-square 28.3%

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

       4. associate-*l* 28.3%

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

   14. Simplified 28.3%

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

   if 9.9999999999999993e260 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 5.5%

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

   3. Taylor expanded in F around 0 7.2%

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

   5. Simplified 14.4%

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

   6. Taylor expanded in B around inf 28.6%

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

      1. pow1/2 28.6%

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

      2. *-commutative 28.6%

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

      3. unpow-prod-down 40.8%

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

      4. pow1/2 40.8%

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

      5. pow1/2 40.8%

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

   8. Applied egg-rr 40.8%

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

4. Final simplification 31.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{+261}:\\ \;\;\;\;-\sqrt{\left|-0.5 \cdot \left(2 \cdot \frac{F}{A}\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{\frac{1 + \left(\frac{A}{B} + \frac{C}{B}\right)}{B}}\right) \cdot \left(-\sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 43.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 10^{+260}:\\ \;\;\;\;-\sqrt{\left|-0.5 \cdot \left(2 \cdot \frac{F}{A}\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2}{B\_m}} \cdot \left(-\sqrt{F}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 1e+260)
   (- (sqrt (fabs (* -0.5 (* 2.0 (/ F A))))))
   (* (sqrt (/ 2.0 B_m)) (- (sqrt F)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 1e+260) {
		tmp = -sqrt(fabs((-0.5 * (2.0 * (F / A)))));
	} else {
		tmp = sqrt((2.0 / B_m)) * -sqrt(F);
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
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 ((b_m ** 2.0d0) <= 1d+260) then
        tmp = -sqrt(abs(((-0.5d0) * (2.0d0 * (f / a)))))
    else
        tmp = sqrt((2.0d0 / b_m)) * -sqrt(f)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (Math.pow(B_m, 2.0) <= 1e+260) {
		tmp = -Math.sqrt(Math.abs((-0.5 * (2.0 * (F / A)))));
	} else {
		tmp = Math.sqrt((2.0 / B_m)) * -Math.sqrt(F);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if math.pow(B_m, 2.0) <= 1e+260:
		tmp = -math.sqrt(math.fabs((-0.5 * (2.0 * (F / A)))))
	else:
		tmp = math.sqrt((2.0 / B_m)) * -math.sqrt(F)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e+260)
		tmp = Float64(-sqrt(abs(Float64(-0.5 * Float64(2.0 * Float64(F / A))))));
	else
		tmp = Float64(sqrt(Float64(2.0 / B_m)) * Float64(-sqrt(F)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 1e+260)
		tmp = -sqrt(abs((-0.5 * (2.0 * (F / A)))));
	else
		tmp = sqrt((2.0 / B_m)) * -sqrt(F);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+260], (-N[Sqrt[N[Abs[N[(-0.5 * N[(2.0 * N[(F / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), N[(N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.00000000000000007e260

   1. Initial program 25.1%

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

   3. Taylor expanded in F around 0 24.6%

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

   5. Simplified 41.6%

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

   6. Taylor expanded in C around inf 26.3%

      \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{-0.5 \cdot \frac{F}{A}}} \]
   7. Step-by-step derivation

      1. *-commutative 26.3%

         \[\leadsto -\color{blue}{\sqrt{-0.5 \cdot \frac{F}{A}} \cdot \sqrt{2}} \]

      2. pow1/2 26.5%

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

      3. pow1/2 26.5%

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

      4. pow-prod-down 26.6%

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

      5. associate-*r/ 26.6%

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

   8. Applied egg-rr 26.6%

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

      1. unpow1/2 26.4%

         \[\leadsto -\color{blue}{\sqrt{\frac{-0.5 \cdot F}{A} \cdot 2}} \]

   10. Simplified 26.4%

       \[\leadsto -\color{blue}{\sqrt{\frac{-0.5 \cdot F}{A} \cdot 2}} \]
   11. Step-by-step derivation

       1. add-sqr-sqrt 26.4%

          \[\leadsto -\sqrt{\color{blue}{\sqrt{\frac{-0.5 \cdot F}{A} \cdot 2} \cdot \sqrt{\frac{-0.5 \cdot F}{A} \cdot 2}}} \]

       2. pow1/2 26.4%

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

       3. pow1/2 26.6%

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

       4. pow-prod-down 23.8%

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

       5. pow2 23.8%

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

       6. associate-/l* 23.8%

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

   12. Applied egg-rr 23.8%

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

       1. unpow1/2 23.8%

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

       2. unpow2 23.8%

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

       3. rem-sqrt-square 28.4%

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

       4. associate-*l* 28.4%

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

   14. Simplified 28.4%

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

   if 1.00000000000000007e260 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 5.4%

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

   3. Taylor expanded in B around inf 26.7%

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

      1. mul-1-neg 26.7%

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

      2. *-commutative 26.7%

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

   5. Simplified 26.7%

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

      1. *-commutative 26.7%

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

      2. pow1/2 26.7%

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

      3. pow1/2 26.7%

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

      4. pow-prod-down 26.9%

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

   7. Applied egg-rr 26.9%

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

      1. unpow1/2 26.9%

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

   9. Simplified 26.9%

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

       1. associate-*l/ 26.9%

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

   11. Applied egg-rr 26.9%

       \[\leadsto -\sqrt{\color{blue}{\frac{F \cdot 2}{B}}} \]
   12. Step-by-step derivation

       1. associate-/l* 26.9%

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

   13. Simplified 26.9%

       \[\leadsto -\sqrt{\color{blue}{F \cdot \frac{2}{B}}} \]
   14. Step-by-step derivation

       1. pow1/2 26.9%

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

       2. *-commutative 26.9%

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

       3. unpow-prod-down 38.4%

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

       4. pow1/2 38.4%

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

       5. pow1/2 38.4%

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

   15. Applied egg-rr 38.4%

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

4. Final simplification 30.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{+260}:\\ \;\;\;\;-\sqrt{\left|-0.5 \cdot \left(2 \cdot \frac{F}{A}\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2}{B}} \cdot \left(-\sqrt{F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 42.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 7.5 \cdot 10^{+129}:\\ \;\;\;\;-\sqrt{\frac{-F}{A}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2}{B\_m}} \cdot \left(-\sqrt{F}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 7.5e+129)
   (- (sqrt (/ (- F) A)))
   (* (sqrt (/ 2.0 B_m)) (- (sqrt F)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 7.5e+129) {
		tmp = -sqrt((-F / A));
	} else {
		tmp = sqrt((2.0 / B_m)) * -sqrt(F);
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
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 (b_m <= 7.5d+129) then
        tmp = -sqrt((-f / a))
    else
        tmp = sqrt((2.0d0 / b_m)) * -sqrt(f)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 7.5e+129) {
		tmp = -Math.sqrt((-F / A));
	} else {
		tmp = Math.sqrt((2.0 / B_m)) * -Math.sqrt(F);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 7.5e+129:
		tmp = -math.sqrt((-F / A))
	else:
		tmp = math.sqrt((2.0 / B_m)) * -math.sqrt(F)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 7.5e+129)
		tmp = Float64(-sqrt(Float64(Float64(-F) / A)));
	else
		tmp = Float64(sqrt(Float64(2.0 / B_m)) * Float64(-sqrt(F)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 7.5e+129)
		tmp = -sqrt((-F / A));
	else
		tmp = sqrt((2.0 / B_m)) * -sqrt(F);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 7.5e+129], (-N[Sqrt[N[((-F) / A), $MachinePrecision]], $MachinePrecision]), N[(N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 7.4999999999999998e129

   1. Initial program 22.9%

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

   3. Taylor expanded in F around 0 22.4%

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

   5. Simplified 38.2%

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

   6. Taylor expanded in C around inf 23.9%

      \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{-0.5 \cdot \frac{F}{A}}} \]
   7. Step-by-step derivation

      1. *-commutative 23.9%

         \[\leadsto -\color{blue}{\sqrt{-0.5 \cdot \frac{F}{A}} \cdot \sqrt{2}} \]

      2. pow1/2 24.1%

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

      3. pow1/2 24.1%

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

      4. pow-prod-down 24.2%

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

      5. associate-*r/ 24.2%

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

   8. Applied egg-rr 24.2%

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

      1. unpow1/2 24.0%

         \[\leadsto -\color{blue}{\sqrt{\frac{-0.5 \cdot F}{A} \cdot 2}} \]

   10. Simplified 24.0%

       \[\leadsto -\color{blue}{\sqrt{\frac{-0.5 \cdot F}{A} \cdot 2}} \]

   11. Taylor expanded in F around 0 24.0%

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

       1. associate-*r/ 24.0%

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

       2. mul-1-neg 24.0%

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

   13. Simplified 24.0%

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

   if 7.4999999999999998e129 < B

   1. Initial program 3.8%

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

   3. Taylor expanded in B around inf 50.1%

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

      1. mul-1-neg 50.1%

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

      2. *-commutative 50.1%

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

   5. Simplified 50.1%

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

      1. *-commutative 50.1%

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

      2. pow1/2 50.1%

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

      3. pow1/2 50.1%

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

      4. pow-prod-down 50.4%

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

   7. Applied egg-rr 50.4%

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

      1. unpow1/2 50.4%

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

   9. Simplified 50.4%

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

       1. associate-*l/ 50.4%

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

   11. Applied egg-rr 50.4%

       \[\leadsto -\sqrt{\color{blue}{\frac{F \cdot 2}{B}}} \]
   12. Step-by-step derivation

       1. associate-/l* 50.4%

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

   13. Simplified 50.4%

       \[\leadsto -\sqrt{\color{blue}{F \cdot \frac{2}{B}}} \]
   14. Step-by-step derivation

       1. pow1/2 50.4%

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

       2. *-commutative 50.4%

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

       3. unpow-prod-down 74.2%

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

       4. pow1/2 74.2%

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

       5. pow1/2 74.2%

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

   15. Applied egg-rr 74.2%

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

4. Final simplification 29.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 7.5 \cdot 10^{+129}:\\ \;\;\;\;-\sqrt{\frac{-F}{A}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2}{B}} \cdot \left(-\sqrt{F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 36.4% accurate, 5.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 1.32 \cdot 10^{+132}:\\ \;\;\;\;-\sqrt{\frac{-F}{A}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{F \cdot \left(2 \cdot \frac{1 + \left(\frac{A}{B\_m} + \frac{C}{B\_m}\right)}{B\_m}\right)}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 1.32e+132)
   (- (sqrt (/ (- F) A)))
   (- (sqrt (* F (* 2.0 (/ (+ 1.0 (+ (/ A B_m) (/ C B_m))) B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.32e+132) {
		tmp = -sqrt((-F / A));
	} else {
		tmp = -sqrt((F * (2.0 * ((1.0 + ((A / B_m) + (C / B_m))) / B_m))));
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
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 (b_m <= 1.32d+132) then
        tmp = -sqrt((-f / a))
    else
        tmp = -sqrt((f * (2.0d0 * ((1.0d0 + ((a / b_m) + (c / b_m))) / b_m))))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.32e+132) {
		tmp = -Math.sqrt((-F / A));
	} else {
		tmp = -Math.sqrt((F * (2.0 * ((1.0 + ((A / B_m) + (C / B_m))) / B_m))));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 1.32e+132:
		tmp = -math.sqrt((-F / A))
	else:
		tmp = -math.sqrt((F * (2.0 * ((1.0 + ((A / B_m) + (C / B_m))) / B_m))))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 1.32e+132)
		tmp = Float64(-sqrt(Float64(Float64(-F) / A)));
	else
		tmp = Float64(-sqrt(Float64(F * Float64(2.0 * Float64(Float64(1.0 + Float64(Float64(A / B_m) + Float64(C / B_m))) / B_m)))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 1.32e+132)
		tmp = -sqrt((-F / A));
	else
		tmp = -sqrt((F * (2.0 * ((1.0 + ((A / B_m) + (C / B_m))) / B_m))));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 1.32e+132], (-N[Sqrt[N[((-F) / A), $MachinePrecision]], $MachinePrecision]), (-N[Sqrt[N[(F * N[(2.0 * N[(N[(1.0 + N[(N[(A / B$95$m), $MachinePrecision] + N[(C / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if B < 1.3199999999999999e132

   1. Initial program 22.8%

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

   3. Taylor expanded in F around 0 22.3%

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

   5. Simplified 38.1%

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

   6. Taylor expanded in C around inf 23.8%

      \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{-0.5 \cdot \frac{F}{A}}} \]
   7. Step-by-step derivation

      1. *-commutative 23.8%

         \[\leadsto -\color{blue}{\sqrt{-0.5 \cdot \frac{F}{A}} \cdot \sqrt{2}} \]

      2. pow1/2 24.0%

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

      3. pow1/2 24.0%

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

      4. pow-prod-down 24.1%

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

      5. associate-*r/ 24.1%

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

   8. Applied egg-rr 24.1%

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

      1. unpow1/2 23.9%

         \[\leadsto -\color{blue}{\sqrt{\frac{-0.5 \cdot F}{A} \cdot 2}} \]

   10. Simplified 23.9%

       \[\leadsto -\color{blue}{\sqrt{\frac{-0.5 \cdot F}{A} \cdot 2}} \]

   11. Taylor expanded in F around 0 23.9%

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

       1. associate-*r/ 23.9%

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

       2. mul-1-neg 23.9%

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

   13. Simplified 23.9%

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

   if 1.3199999999999999e132 < B

   1. Initial program 3.8%

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

   3. Taylor expanded in F around 0 7.2%

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

   5. Simplified 14.2%

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

   6. Taylor expanded in B around inf 51.6%

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

      1. pow1 51.6%

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

      2. sqrt-unprod 51.9%

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

   8. Applied egg-rr 51.9%

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

      1. unpow1 51.9%

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

      2. *-commutative 51.9%

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

      3. associate-*l* 51.9%

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

      4. +-commutative 51.9%

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

   10. Simplified 51.9%

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

4. Final simplification 27.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.32 \cdot 10^{+132}:\\ \;\;\;\;-\sqrt{\frac{-F}{A}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{F \cdot \left(2 \cdot \frac{1 + \left(\frac{A}{B} + \frac{C}{B}\right)}{B}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 36.4% accurate, 5.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 1.7 \cdot 10^{+130}:\\ \;\;\;\;-\sqrt{\frac{-F}{A}}\\ \mathbf{else}:\\ \;\;\;\;-{\left(\frac{2 \cdot F}{B\_m}\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 1.7e+130) (- (sqrt (/ (- F) A))) (- (pow (/ (* 2.0 F) B_m) 0.5))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.7e+130) {
		tmp = -sqrt((-F / A));
	} else {
		tmp = -pow(((2.0 * F) / B_m), 0.5);
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
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 (b_m <= 1.7d+130) then
        tmp = -sqrt((-f / a))
    else
        tmp = -(((2.0d0 * f) / b_m) ** 0.5d0)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.7e+130) {
		tmp = -Math.sqrt((-F / A));
	} else {
		tmp = -Math.pow(((2.0 * F) / B_m), 0.5);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 1.7e+130:
		tmp = -math.sqrt((-F / A))
	else:
		tmp = -math.pow(((2.0 * F) / B_m), 0.5)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 1.7e+130)
		tmp = Float64(-sqrt(Float64(Float64(-F) / A)));
	else
		tmp = Float64(-(Float64(Float64(2.0 * F) / B_m) ^ 0.5));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 1.7e+130)
		tmp = -sqrt((-F / A));
	else
		tmp = -(((2.0 * F) / B_m) ^ 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 1.7e+130], (-N[Sqrt[N[((-F) / A), $MachinePrecision]], $MachinePrecision]), (-N[Power[N[(N[(2.0 * F), $MachinePrecision] / B$95$m), $MachinePrecision], 0.5], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if B < 1.7e130

   1. Initial program 22.9%

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

   3. Taylor expanded in F around 0 22.4%

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

   5. Simplified 38.2%

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

   6. Taylor expanded in C around inf 23.9%

      \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{-0.5 \cdot \frac{F}{A}}} \]
   7. Step-by-step derivation

      1. *-commutative 23.9%

         \[\leadsto -\color{blue}{\sqrt{-0.5 \cdot \frac{F}{A}} \cdot \sqrt{2}} \]

      2. pow1/2 24.1%

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

      3. pow1/2 24.1%

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

      4. pow-prod-down 24.2%

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

      5. associate-*r/ 24.2%

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

   8. Applied egg-rr 24.2%

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

      1. unpow1/2 24.0%

         \[\leadsto -\color{blue}{\sqrt{\frac{-0.5 \cdot F}{A} \cdot 2}} \]

   10. Simplified 24.0%

       \[\leadsto -\color{blue}{\sqrt{\frac{-0.5 \cdot F}{A} \cdot 2}} \]

   11. Taylor expanded in F around 0 24.0%

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

       1. associate-*r/ 24.0%

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

       2. mul-1-neg 24.0%

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

   13. Simplified 24.0%

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

   if 1.7e130 < B

   1. Initial program 3.8%

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

   3. Taylor expanded in B around inf 50.1%

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

      1. mul-1-neg 50.1%

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

      2. *-commutative 50.1%

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

   5. Simplified 50.1%

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

      1. *-commutative 50.1%

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

      2. pow1/2 50.1%

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

      3. pow1/2 50.1%

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

      4. pow-prod-down 50.4%

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

   7. Applied egg-rr 50.4%

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

      1. unpow1/2 50.4%

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

   9. Simplified 50.4%

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

       1. associate-*l/ 50.4%

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

   11. Applied egg-rr 50.4%

       \[\leadsto -\sqrt{\color{blue}{\frac{F \cdot 2}{B}}} \]
   12. Step-by-step derivation

       1. associate-/l* 50.4%

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

   13. Simplified 50.4%

       \[\leadsto -\sqrt{\color{blue}{F \cdot \frac{2}{B}}} \]
   14. Step-by-step derivation

       1. pow1/2 50.4%

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

       2. associate-*r/ 50.4%

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

   15. Applied egg-rr 50.4%

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

4. Final simplification 27.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.7 \cdot 10^{+130}:\\ \;\;\;\;-\sqrt{\frac{-F}{A}}\\ \mathbf{else}:\\ \;\;\;\;-{\left(\frac{2 \cdot F}{B}\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 36.4% accurate, 5.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 7.5 \cdot 10^{+129}:\\ \;\;\;\;-\sqrt{\frac{-F}{A}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{F}{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 7.5e+129) (- (sqrt (/ (- F) A))) (- (sqrt (* 2.0 (/ F B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 7.5e+129) {
		tmp = -sqrt((-F / A));
	} else {
		tmp = -sqrt((2.0 * (F / B_m)));
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
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 (b_m <= 7.5d+129) then
        tmp = -sqrt((-f / a))
    else
        tmp = -sqrt((2.0d0 * (f / b_m)))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 7.5e+129) {
		tmp = -Math.sqrt((-F / A));
	} else {
		tmp = -Math.sqrt((2.0 * (F / B_m)));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 7.5e+129:
		tmp = -math.sqrt((-F / A))
	else:
		tmp = -math.sqrt((2.0 * (F / B_m)))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 7.5e+129)
		tmp = Float64(-sqrt(Float64(Float64(-F) / A)));
	else
		tmp = Float64(-sqrt(Float64(2.0 * Float64(F / B_m))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 7.5e+129)
		tmp = -sqrt((-F / A));
	else
		tmp = -sqrt((2.0 * (F / B_m)));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 7.5e+129], (-N[Sqrt[N[((-F) / A), $MachinePrecision]], $MachinePrecision]), (-N[Sqrt[N[(2.0 * N[(F / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if B < 7.4999999999999998e129

   1. Initial program 22.9%

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

   3. Taylor expanded in F around 0 22.4%

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

   5. Simplified 38.2%

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

   6. Taylor expanded in C around inf 23.9%

      \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{-0.5 \cdot \frac{F}{A}}} \]
   7. Step-by-step derivation

      1. *-commutative 23.9%

         \[\leadsto -\color{blue}{\sqrt{-0.5 \cdot \frac{F}{A}} \cdot \sqrt{2}} \]

      2. pow1/2 24.1%

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

      3. pow1/2 24.1%

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

      4. pow-prod-down 24.2%

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

      5. associate-*r/ 24.2%

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

   8. Applied egg-rr 24.2%

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

      1. unpow1/2 24.0%

         \[\leadsto -\color{blue}{\sqrt{\frac{-0.5 \cdot F}{A} \cdot 2}} \]

   10. Simplified 24.0%

       \[\leadsto -\color{blue}{\sqrt{\frac{-0.5 \cdot F}{A} \cdot 2}} \]

   11. Taylor expanded in F around 0 24.0%

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

       1. associate-*r/ 24.0%

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

       2. mul-1-neg 24.0%

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

   13. Simplified 24.0%

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

   if 7.4999999999999998e129 < B

   1. Initial program 3.8%

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

   3. Taylor expanded in B around inf 50.1%

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

      1. mul-1-neg 50.1%

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

      2. *-commutative 50.1%

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

   5. Simplified 50.1%

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

      1. *-commutative 50.1%

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

      2. pow1/2 50.1%

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

      3. pow1/2 50.1%

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

      4. pow-prod-down 50.4%

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

   7. Applied egg-rr 50.4%

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

      1. unpow1/2 50.4%

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

   9. Simplified 50.4%

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

4. Final simplification 27.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 7.5 \cdot 10^{+129}:\\ \;\;\;\;-\sqrt{\frac{-F}{A}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{F}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 36.4% accurate, 5.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 9 \cdot 10^{+129}:\\ \;\;\;\;-\sqrt{\frac{-F}{A}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{F \cdot \frac{2}{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 9e+129) (- (sqrt (/ (- F) A))) (- (sqrt (* F (/ 2.0 B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 9e+129) {
		tmp = -sqrt((-F / A));
	} else {
		tmp = -sqrt((F * (2.0 / B_m)));
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
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 (b_m <= 9d+129) then
        tmp = -sqrt((-f / a))
    else
        tmp = -sqrt((f * (2.0d0 / b_m)))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 9e+129) {
		tmp = -Math.sqrt((-F / A));
	} else {
		tmp = -Math.sqrt((F * (2.0 / B_m)));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 9e+129:
		tmp = -math.sqrt((-F / A))
	else:
		tmp = -math.sqrt((F * (2.0 / B_m)))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 9e+129)
		tmp = Float64(-sqrt(Float64(Float64(-F) / A)));
	else
		tmp = Float64(-sqrt(Float64(F * Float64(2.0 / B_m))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 9e+129)
		tmp = -sqrt((-F / A));
	else
		tmp = -sqrt((F * (2.0 / B_m)));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 9e+129], (-N[Sqrt[N[((-F) / A), $MachinePrecision]], $MachinePrecision]), (-N[Sqrt[N[(F * N[(2.0 / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if B < 9.0000000000000003e129

   1. Initial program 22.9%

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

   3. Taylor expanded in F around 0 22.4%

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

   5. Simplified 38.2%

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

   6. Taylor expanded in C around inf 23.9%

      \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{-0.5 \cdot \frac{F}{A}}} \]
   7. Step-by-step derivation

      1. *-commutative 23.9%

         \[\leadsto -\color{blue}{\sqrt{-0.5 \cdot \frac{F}{A}} \cdot \sqrt{2}} \]

      2. pow1/2 24.1%

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

      3. pow1/2 24.1%

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

      4. pow-prod-down 24.2%

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

      5. associate-*r/ 24.2%

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

   8. Applied egg-rr 24.2%

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

      1. unpow1/2 24.0%

         \[\leadsto -\color{blue}{\sqrt{\frac{-0.5 \cdot F}{A} \cdot 2}} \]

   10. Simplified 24.0%

       \[\leadsto -\color{blue}{\sqrt{\frac{-0.5 \cdot F}{A} \cdot 2}} \]

   11. Taylor expanded in F around 0 24.0%

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

       1. associate-*r/ 24.0%

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

       2. mul-1-neg 24.0%

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

   13. Simplified 24.0%

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

   if 9.0000000000000003e129 < B

   1. Initial program 3.8%

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

   3. Taylor expanded in B around inf 50.1%

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

      1. mul-1-neg 50.1%

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

      2. *-commutative 50.1%

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

   5. Simplified 50.1%

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

      1. *-commutative 50.1%

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

      2. pow1/2 50.1%

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

      3. pow1/2 50.1%

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

      4. pow-prod-down 50.4%

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

   7. Applied egg-rr 50.4%

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

      1. unpow1/2 50.4%

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

   9. Simplified 50.4%

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

       1. associate-*l/ 50.4%

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

   11. Applied egg-rr 50.4%

       \[\leadsto -\sqrt{\color{blue}{\frac{F \cdot 2}{B}}} \]
   12. Step-by-step derivation

       1. associate-/l* 50.4%

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

   13. Simplified 50.4%

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

Alternative 12: 26.4% accurate, 6.0× speedup

Math

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

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (- (sqrt (/ (- F) A))))

C

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

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
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);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	return -Math.sqrt((-F / A));
}

Python

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

Julia

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

MATLAB

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

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := (-N[Sqrt[N[((-F) / A), $MachinePrecision]], $MachinePrecision])

Derivation

1. Initial program 20.7%

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

3. Taylor expanded in F around 0 20.6%

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

5. Simplified 35.4%

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

6. Taylor expanded in C around inf 21.4%

   \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{-0.5 \cdot \frac{F}{A}}} \]
7. Step-by-step derivation

   1. *-commutative 21.4%

      \[\leadsto -\color{blue}{\sqrt{-0.5 \cdot \frac{F}{A}} \cdot \sqrt{2}} \]

   2. pow1/2 21.5%

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

   3. pow1/2 21.5%

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

   4. pow-prod-down 21.6%

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

   5. associate-*r/ 21.6%

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

8. Applied egg-rr 21.6%

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

   1. unpow1/2 21.5%

      \[\leadsto -\color{blue}{\sqrt{\frac{-0.5 \cdot F}{A} \cdot 2}} \]

10. Simplified 21.5%

    \[\leadsto -\color{blue}{\sqrt{\frac{-0.5 \cdot F}{A} \cdot 2}} \]

11. Taylor expanded in F around 0 21.5%

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

    1. associate-*r/ 21.5%

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

    2. mul-1-neg 21.5%

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

13. Simplified 21.5%

    \[\leadsto -\sqrt{\color{blue}{\frac{-F}{A}}} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024181 
(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))))