ABCF->ab-angle a

Percentage Accurate: 18.7% → 51.7%

Time: 24.2s

Alternatives: 13

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: 18.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 51.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-83}:\\ \;\;\;\;\frac{\left(\sqrt{F} \cdot \sqrt{4 \cdot \mathsf{fma}\left(-4, C \cdot A, {B\_m}^{2}\right)}\right) \cdot \sqrt{C}}{-\mathsf{fma}\left(B\_m, B\_m, A \cdot \left(-4 \cdot C\right)\right)}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+304}:\\ \;\;\;\;\sqrt{F \cdot \frac{A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)}{{B\_m}^{2} + -4 \cdot \left(C \cdot A\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(-\sqrt{\frac{2}{B\_m}}\right)\\ \end{array} \end{array} \]

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 (<= (pow B_m 2.0) 5e-83)
   (/
    (* (* (sqrt F) (sqrt (* 4.0 (fma -4.0 (* C A) (pow B_m 2.0))))) (sqrt C))
    (- (fma B_m B_m (* A (* -4.0 C)))))
   (if (<= (pow B_m 2.0) 5e+304)
     (*
      (sqrt
       (*
        F
        (/
         (+ A (+ C (hypot B_m (- A C))))
         (+ (pow B_m 2.0) (* -4.0 (* C A))))))
      (- (sqrt 2.0)))
     (* (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) <= 5e-83) {
		tmp = ((sqrt(F) * sqrt((4.0 * fma(-4.0, (C * A), pow(B_m, 2.0))))) * sqrt(C)) / -fma(B_m, B_m, (A * (-4.0 * C)));
	} else if (pow(B_m, 2.0) <= 5e+304) {
		tmp = sqrt((F * ((A + (C + hypot(B_m, (A - C)))) / (pow(B_m, 2.0) + (-4.0 * (C * A)))))) * -sqrt(2.0);
	} else {
		tmp = 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) <= 5e-83)
		tmp = Float64(Float64(Float64(sqrt(F) * sqrt(Float64(4.0 * fma(-4.0, Float64(C * A), (B_m ^ 2.0))))) * sqrt(C)) / Float64(-fma(B_m, B_m, Float64(A * Float64(-4.0 * C)))));
	elseif ((B_m ^ 2.0) <= 5e+304)
		tmp = Float64(sqrt(Float64(F * Float64(Float64(A + Float64(C + hypot(B_m, Float64(A - C)))) / Float64((B_m ^ 2.0) + Float64(-4.0 * Float64(C * A)))))) * Float64(-sqrt(2.0)));
	else
		tmp = Float64(sqrt(F) * Float64(-sqrt(Float64(2.0 / 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], 5e-83], N[(N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(4.0 * N[(-4.0 * N[(C * A), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[C], $MachinePrecision]), $MachinePrecision] / (-N[(B$95$m * B$95$m + N[(A * N[(-4.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+304], N[(N[Sqrt[N[(F * N[(N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[(-4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 5e-83

   1. Initial program 17.7%

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

   3. Simplified 23.3%

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

   5. Taylor expanded in A around -inf 16.8%

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

      1. pow1/2 16.9%

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

      2. associate-*r* 16.9%

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

      3. unpow-prod-down 20.9%

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

      4. *-commutative 20.9%

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

      5. fma-undefine 20.9%

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

      6. unpow2 20.9%

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

      7. associate-*r* 20.9%

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

      8. *-commutative 20.9%

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

      9. +-commutative 20.9%

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

      10. *-commutative 20.9%

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

      11. associate-*r* 20.9%

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

      12. fma-define 20.9%

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

      13. pow1/2 20.9%

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

   7. Applied egg-rr 20.9%

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

      1. unpow1/2 20.9%

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

   9. Simplified 20.9%

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

       1. pow1/2 20.9%

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

       2. associate-*l* 20.9%

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

       3. unpow-prod-down 27.8%

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

       4. pow1/2 27.8%

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

   11. Applied egg-rr 27.8%

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

       1. unpow1/2 27.8%

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

       2. *-commutative 27.8%

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

       3. fma-undefine 27.8%

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

       4. associate-*r* 27.8%

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

       5. *-commutative 27.8%

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

       6. fma-define 27.8%

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

       7. *-commutative 27.8%

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

   13. Simplified 27.8%

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

   if 5e-83 < (pow.f64 B #s(literal 2 binary64)) < 4.9999999999999997e304

   1. Initial program 30.9%

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

   3. Taylor expanded in F around 0 36.9%

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

      1. mul-1-neg 36.9%

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

   5. Simplified 53.4%

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

   if 4.9999999999999997e304 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 0.1%

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

   3. Taylor expanded in B around inf 31.1%

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

      1. mul-1-neg 31.1%

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

   5. Simplified 31.1%

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

      1. sqrt-div 43.8%

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

   7. Applied egg-rr 43.8%

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

      1. associate-*l/ 44.0%

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

      2. pow1/2 44.0%

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

      3. pow1/2 44.0%

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

      4. pow-prod-down 44.0%

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

   9. Applied egg-rr 44.0%

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

       1. unpow1/2 44.0%

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

   11. Simplified 44.0%

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

       1. sqrt-undiv 31.4%

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

       2. associate-*r/ 31.3%

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

       3. pow1/2 31.3%

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

       4. *-commutative 31.3%

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

       5. unpow-prod-down 44.0%

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

       6. pow1/2 44.0%

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

       7. pow1/2 44.0%

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

   13. Applied egg-rr 44.0%

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

4. Final simplification 40.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-83}:\\ \;\;\;\;\frac{\left(\sqrt{F} \cdot \sqrt{4 \cdot \mathsf{fma}\left(-4, C \cdot A, {B}^{2}\right)}\right) \cdot \sqrt{C}}{-\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+304}:\\ \;\;\;\;\sqrt{F \cdot \frac{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}{{B}^{2} + -4 \cdot \left(C \cdot A\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(-\sqrt{\frac{2}{B}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 52.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := -\sqrt{2}\\ \mathbf{if}\;{B\_m}^{2} \leq 4 \cdot 10^{-89}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{A}} \cdot t\_0\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+304}:\\ \;\;\;\;\sqrt{F \cdot \frac{A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)}{{B\_m}^{2} + -4 \cdot \left(C \cdot A\right)}} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(-\sqrt{\frac{2}{B\_m}}\right)\\ \end{array} \end{array} \]

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
 (let* ((t_0 (- (sqrt 2.0))))
   (if (<= (pow B_m 2.0) 4e-89)
     (* (sqrt (* F (/ -0.5 A))) t_0)
     (if (<= (pow B_m 2.0) 5e+304)
       (*
        (sqrt
         (*
          F
          (/
           (+ A (+ C (hypot B_m (- A C))))
           (+ (pow B_m 2.0) (* -4.0 (* C A))))))
        t_0)
       (* (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 = -sqrt(2.0);
	double tmp;
	if (pow(B_m, 2.0) <= 4e-89) {
		tmp = sqrt((F * (-0.5 / A))) * t_0;
	} else if (pow(B_m, 2.0) <= 5e+304) {
		tmp = sqrt((F * ((A + (C + hypot(B_m, (A - C)))) / (pow(B_m, 2.0) + (-4.0 * (C * A)))))) * t_0;
	} else {
		tmp = 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 t_0 = -Math.sqrt(2.0);
	double tmp;
	if (Math.pow(B_m, 2.0) <= 4e-89) {
		tmp = Math.sqrt((F * (-0.5 / A))) * t_0;
	} else if (Math.pow(B_m, 2.0) <= 5e+304) {
		tmp = Math.sqrt((F * ((A + (C + Math.hypot(B_m, (A - C)))) / (Math.pow(B_m, 2.0) + (-4.0 * (C * A)))))) * t_0;
	} else {
		tmp = 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):
	t_0 = -math.sqrt(2.0)
	tmp = 0
	if math.pow(B_m, 2.0) <= 4e-89:
		tmp = math.sqrt((F * (-0.5 / A))) * t_0
	elif math.pow(B_m, 2.0) <= 5e+304:
		tmp = math.sqrt((F * ((A + (C + math.hypot(B_m, (A - C)))) / (math.pow(B_m, 2.0) + (-4.0 * (C * A)))))) * t_0
	else:
		tmp = 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)
	t_0 = Float64(-sqrt(2.0))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 4e-89)
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 / A))) * t_0);
	elseif ((B_m ^ 2.0) <= 5e+304)
		tmp = Float64(sqrt(Float64(F * Float64(Float64(A + Float64(C + hypot(B_m, Float64(A - C)))) / Float64((B_m ^ 2.0) + Float64(-4.0 * Float64(C * A)))))) * t_0);
	else
		tmp = Float64(sqrt(F) * Float64(-sqrt(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)
	t_0 = -sqrt(2.0);
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 4e-89)
		tmp = sqrt((F * (-0.5 / A))) * t_0;
	elseif ((B_m ^ 2.0) <= 5e+304)
		tmp = sqrt((F * ((A + (C + hypot(B_m, (A - C)))) / ((B_m ^ 2.0) + (-4.0 * (C * A)))))) * t_0;
	else
		tmp = 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_] := Block[{t$95$0 = (-N[Sqrt[2.0], $MachinePrecision])}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e-89], N[(N[Sqrt[N[(F * N[(-0.5 / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+304], N[(N[Sqrt[N[(F * N[(N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[(-4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 4.00000000000000015e-89

   1. Initial program 17.4%

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

   3. Taylor expanded in F around 0 17.4%

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

      1. mul-1-neg 17.4%

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

   5. Simplified 29.2%

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

   6. Taylor expanded in A around -inf 29.5%

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

   if 4.00000000000000015e-89 < (pow.f64 B #s(literal 2 binary64)) < 4.9999999999999997e304

   1. Initial program 30.7%

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

   3. Taylor expanded in F around 0 36.4%

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

      1. mul-1-neg 36.4%

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

   5. Simplified 54.5%

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

   if 4.9999999999999997e304 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 0.1%

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

   3. Taylor expanded in B around inf 31.1%

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

      1. mul-1-neg 31.1%

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

   5. Simplified 31.1%

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

      1. sqrt-div 43.8%

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

   7. Applied egg-rr 43.8%

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

      1. associate-*l/ 44.0%

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

      2. pow1/2 44.0%

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

      3. pow1/2 44.0%

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

      4. pow-prod-down 44.0%

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

   9. Applied egg-rr 44.0%

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

       1. unpow1/2 44.0%

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

   11. Simplified 44.0%

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

       1. sqrt-undiv 31.4%

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

       2. associate-*r/ 31.3%

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

       3. pow1/2 31.3%

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

       4. *-commutative 31.3%

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

       5. unpow-prod-down 44.0%

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

       6. pow1/2 44.0%

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

       7. pow1/2 44.0%

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

   13. Applied egg-rr 44.0%

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

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 4 \cdot 10^{-89}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+304}:\\ \;\;\;\;\sqrt{F \cdot \frac{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}{{B}^{2} + -4 \cdot \left(C \cdot A\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(-\sqrt{\frac{2}{B}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 50.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := -\sqrt{2}\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{-82}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{A}} \cdot t\_0\\ \mathbf{elif}\;{B\_m}^{2} \leq 2.5 \cdot 10^{+259}:\\ \;\;\;\;\sqrt{F \cdot \frac{C + \mathsf{hypot}\left(B\_m, C\right)}{{B\_m}^{2}}} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(-\sqrt{\frac{2}{B\_m}}\right)\\ \end{array} \end{array} \]

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
 (let* ((t_0 (- (sqrt 2.0))))
   (if (<= (pow B_m 2.0) 1e-82)
     (* (sqrt (* F (/ -0.5 A))) t_0)
     (if (<= (pow B_m 2.0) 2.5e+259)
       (* (sqrt (* F (/ (+ C (hypot B_m C)) (pow B_m 2.0)))) t_0)
       (* (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 = -sqrt(2.0);
	double tmp;
	if (pow(B_m, 2.0) <= 1e-82) {
		tmp = sqrt((F * (-0.5 / A))) * t_0;
	} else if (pow(B_m, 2.0) <= 2.5e+259) {
		tmp = sqrt((F * ((C + hypot(B_m, C)) / pow(B_m, 2.0)))) * t_0;
	} else {
		tmp = 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 t_0 = -Math.sqrt(2.0);
	double tmp;
	if (Math.pow(B_m, 2.0) <= 1e-82) {
		tmp = Math.sqrt((F * (-0.5 / A))) * t_0;
	} else if (Math.pow(B_m, 2.0) <= 2.5e+259) {
		tmp = Math.sqrt((F * ((C + Math.hypot(B_m, C)) / Math.pow(B_m, 2.0)))) * t_0;
	} else {
		tmp = 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):
	t_0 = -math.sqrt(2.0)
	tmp = 0
	if math.pow(B_m, 2.0) <= 1e-82:
		tmp = math.sqrt((F * (-0.5 / A))) * t_0
	elif math.pow(B_m, 2.0) <= 2.5e+259:
		tmp = math.sqrt((F * ((C + math.hypot(B_m, C)) / math.pow(B_m, 2.0)))) * t_0
	else:
		tmp = 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)
	t_0 = Float64(-sqrt(2.0))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-82)
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 / A))) * t_0);
	elseif ((B_m ^ 2.0) <= 2.5e+259)
		tmp = Float64(sqrt(Float64(F * Float64(Float64(C + hypot(B_m, C)) / (B_m ^ 2.0)))) * t_0);
	else
		tmp = Float64(sqrt(F) * Float64(-sqrt(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)
	t_0 = -sqrt(2.0);
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 1e-82)
		tmp = sqrt((F * (-0.5 / A))) * t_0;
	elseif ((B_m ^ 2.0) <= 2.5e+259)
		tmp = sqrt((F * ((C + hypot(B_m, C)) / (B_m ^ 2.0)))) * t_0;
	else
		tmp = 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_] := Block[{t$95$0 = (-N[Sqrt[2.0], $MachinePrecision])}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-82], N[(N[Sqrt[N[(F * N[(-0.5 / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2.5e+259], N[(N[Sqrt[N[(F * N[(N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision] / N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1e-82

   1. Initial program 17.6%

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

   3. Taylor expanded in F around 0 17.6%

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

      1. mul-1-neg 17.6%

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

   5. Simplified 30.7%

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

   6. Taylor expanded in A around -inf 30.1%

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

   if 1e-82 < (pow.f64 B #s(literal 2 binary64)) < 2.50000000000000016e259

   1. Initial program 31.9%

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

   3. Taylor expanded in F around 0 39.7%

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

      1. mul-1-neg 39.7%

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

   5. Simplified 53.7%

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

   6. Taylor expanded in A around 0 42.4%

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

      1. unpow2 42.4%

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

      2. unpow2 42.4%

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

      3. hypot-undefine 45.6%

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

   8. Simplified 45.6%

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

   if 2.50000000000000016e259 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 1.5%

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

   3. Taylor expanded in B around inf 31.5%

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

      1. mul-1-neg 31.5%

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

   5. Simplified 31.5%

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

      1. sqrt-div 44.5%

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

   7. Applied egg-rr 44.5%

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

      1. associate-*l/ 44.6%

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

      2. pow1/2 44.6%

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

      3. pow1/2 44.6%

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

      4. pow-prod-down 44.6%

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

   9. Applied egg-rr 44.6%

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

       1. unpow1/2 44.6%

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

   11. Simplified 44.6%

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

       1. sqrt-undiv 31.8%

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

       2. associate-*r/ 31.7%

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

       3. pow1/2 31.7%

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

       4. *-commutative 31.7%

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

       5. unpow-prod-down 44.7%

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

       6. pow1/2 44.7%

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

       7. pow1/2 44.7%

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

   13. Applied egg-rr 44.7%

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

4. Final simplification 38.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-82}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;{B}^{2} \leq 2.5 \cdot 10^{+259}:\\ \;\;\;\;\sqrt{F \cdot \frac{C + \mathsf{hypot}\left(B, C\right)}{{B}^{2}}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(-\sqrt{\frac{2}{B}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 48.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-81}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

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) 5e-81)
   (* (sqrt (* F (/ -0.5 A))) (- (sqrt 2.0)))
   (/ (sqrt (* 2.0 F)) (- (sqrt 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) <= 5e-81) {
		tmp = sqrt((F * (-0.5 / A))) * -sqrt(2.0);
	} else {
		tmp = sqrt((2.0 * F)) / -sqrt(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 ** 2.0d0) <= 5d-81) then
        tmp = sqrt((f * ((-0.5d0) / a))) * -sqrt(2.0d0)
    else
        tmp = sqrt((2.0d0 * f)) / -sqrt(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 (Math.pow(B_m, 2.0) <= 5e-81) {
		tmp = Math.sqrt((F * (-0.5 / A))) * -Math.sqrt(2.0);
	} else {
		tmp = Math.sqrt((2.0 * F)) / -Math.sqrt(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) <= 5e-81:
		tmp = math.sqrt((F * (-0.5 / A))) * -math.sqrt(2.0)
	else:
		tmp = math.sqrt((2.0 * F)) / -math.sqrt(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) <= 5e-81)
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 / A))) * Float64(-sqrt(2.0)));
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) / Float64(-sqrt(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) <= 5e-81)
		tmp = sqrt((F * (-0.5 / A))) * -sqrt(2.0);
	else
		tmp = sqrt((2.0 * F)) / -sqrt(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], 5e-81], N[(N[Sqrt[N[(F * N[(-0.5 / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 4.99999999999999981e-81

   1. Initial program 18.3%

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

   3. Taylor expanded in F around 0 18.3%

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

      1. mul-1-neg 18.3%

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

   5. Simplified 31.4%

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

   6. Taylor expanded in A around -inf 29.8%

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

   if 4.99999999999999981e-81 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 16.1%

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

   3. Taylor expanded in B around inf 27.4%

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

      1. mul-1-neg 27.4%

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

   5. Simplified 27.4%

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

      1. sqrt-div 33.9%

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

   7. Applied egg-rr 33.9%

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

      1. associate-*l/ 33.9%

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

      2. pow1/2 33.9%

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

      3. pow1/2 33.9%

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

      4. pow-prod-down 34.0%

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

   9. Applied egg-rr 34.0%

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

       1. unpow1/2 34.0%

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

   11. Simplified 34.0%

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

4. Final simplification 32.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-81}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 49.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 2.7 \cdot 10^{-40}:\\ \;\;\;\;\frac{\sqrt{\left(\left(-4 \cdot A\right) \cdot \left(F \cdot C\right)\right) \cdot \left(4 \cdot C\right)}}{4 \cdot \left(C \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

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 2.7e-40)
   (/ (sqrt (* (* (* -4.0 A) (* F C)) (* 4.0 C))) (* 4.0 (* C A)))
   (/ (sqrt (* 2.0 F)) (- (sqrt 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 <= 2.7e-40) {
		tmp = sqrt((((-4.0 * A) * (F * C)) * (4.0 * C))) / (4.0 * (C * A));
	} else {
		tmp = sqrt((2.0 * F)) / -sqrt(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 <= 2.7d-40) then
        tmp = sqrt(((((-4.0d0) * a) * (f * c)) * (4.0d0 * c))) / (4.0d0 * (c * a))
    else
        tmp = sqrt((2.0d0 * f)) / -sqrt(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 <= 2.7e-40) {
		tmp = Math.sqrt((((-4.0 * A) * (F * C)) * (4.0 * C))) / (4.0 * (C * A));
	} else {
		tmp = Math.sqrt((2.0 * F)) / -Math.sqrt(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 <= 2.7e-40:
		tmp = math.sqrt((((-4.0 * A) * (F * C)) * (4.0 * C))) / (4.0 * (C * A))
	else:
		tmp = math.sqrt((2.0 * F)) / -math.sqrt(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 <= 2.7e-40)
		tmp = Float64(sqrt(Float64(Float64(Float64(-4.0 * A) * Float64(F * C)) * Float64(4.0 * C))) / Float64(4.0 * Float64(C * A)));
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) / Float64(-sqrt(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 <= 2.7e-40)
		tmp = sqrt((((-4.0 * A) * (F * C)) * (4.0 * C))) / (4.0 * (C * A));
	else
		tmp = sqrt((2.0 * F)) / -sqrt(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, 2.7e-40], N[(N[Sqrt[N[(N[(N[(-4.0 * A), $MachinePrecision] * N[(F * C), $MachinePrecision]), $MachinePrecision] * N[(4.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 2.7e-40

   1. Initial program 18.7%

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

   3. Simplified 24.7%

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

   5. Taylor expanded in A around -inf 12.8%

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

   6. Taylor expanded in B around 0 11.3%

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

   7. Taylor expanded in B around 0 11.9%

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

      1. associate-*r* 12.0%

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

   9. Simplified 12.0%

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

   if 2.7e-40 < B

   1. Initial program 13.7%

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

   3. Taylor expanded in B around inf 47.0%

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

      1. mul-1-neg 47.0%

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

   5. Simplified 47.0%

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

      1. sqrt-div 59.3%

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

   7. Applied egg-rr 59.3%

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

      1. associate-*l/ 59.5%

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

      2. pow1/2 59.5%

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

      3. pow1/2 59.5%

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

      4. pow-prod-down 59.6%

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

   9. Applied egg-rr 59.6%

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

       1. unpow1/2 59.6%

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

   11. Simplified 59.6%

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

4. Final simplification 27.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.7 \cdot 10^{-40}:\\ \;\;\;\;\frac{\sqrt{\left(\left(-4 \cdot A\right) \cdot \left(F \cdot C\right)\right) \cdot \left(4 \cdot C\right)}}{4 \cdot \left(C \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 49.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 2.6 \cdot 10^{-36}:\\ \;\;\;\;\frac{\sqrt{\left(\left(-4 \cdot A\right) \cdot \left(F \cdot C\right)\right) \cdot \left(4 \cdot C\right)}}{4 \cdot \left(C \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(-\sqrt{\frac{2}{B\_m}}\right)\\ \end{array} \end{array} \]

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 2.6e-36)
   (/ (sqrt (* (* (* -4.0 A) (* F C)) (* 4.0 C))) (* 4.0 (* C A)))
   (* (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 (B_m <= 2.6e-36) {
		tmp = sqrt((((-4.0 * A) * (F * C)) * (4.0 * C))) / (4.0 * (C * A));
	} else {
		tmp = sqrt(F) * -sqrt((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 <= 2.6d-36) then
        tmp = sqrt(((((-4.0d0) * a) * (f * c)) * (4.0d0 * c))) / (4.0d0 * (c * a))
    else
        tmp = sqrt(f) * -sqrt((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 <= 2.6e-36) {
		tmp = Math.sqrt((((-4.0 * A) * (F * C)) * (4.0 * C))) / (4.0 * (C * A));
	} else {
		tmp = 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 B_m <= 2.6e-36:
		tmp = math.sqrt((((-4.0 * A) * (F * C)) * (4.0 * C))) / (4.0 * (C * A))
	else:
		tmp = 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 <= 2.6e-36)
		tmp = Float64(sqrt(Float64(Float64(Float64(-4.0 * A) * Float64(F * C)) * Float64(4.0 * C))) / Float64(4.0 * Float64(C * A)));
	else
		tmp = Float64(sqrt(F) * Float64(-sqrt(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 <= 2.6e-36)
		tmp = sqrt((((-4.0 * A) * (F * C)) * (4.0 * C))) / (4.0 * (C * A));
	else
		tmp = 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[B$95$m, 2.6e-36], N[(N[Sqrt[N[(N[(N[(-4.0 * A), $MachinePrecision] * N[(F * C), $MachinePrecision]), $MachinePrecision] * N[(4.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 2.6e-36

   1. Initial program 18.7%

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

   3. Simplified 24.7%

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

   5. Taylor expanded in A around -inf 12.8%

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

   6. Taylor expanded in B around 0 11.3%

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

   7. Taylor expanded in B around 0 11.9%

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

      1. associate-*r* 12.0%

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

   9. Simplified 12.0%

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

   if 2.6e-36 < B

   1. Initial program 13.7%

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

   3. Taylor expanded in B around inf 47.0%

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

      1. mul-1-neg 47.0%

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

   5. Simplified 47.0%

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

      1. sqrt-div 59.3%

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

   7. Applied egg-rr 59.3%

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

      1. associate-*l/ 59.5%

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

      2. pow1/2 59.5%

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

      3. pow1/2 59.5%

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

      4. pow-prod-down 59.6%

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

   9. Applied egg-rr 59.6%

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

       1. unpow1/2 59.6%

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

   11. Simplified 59.6%

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

       1. sqrt-undiv 47.3%

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

       2. associate-*r/ 47.3%

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

       3. pow1/2 47.3%

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

       4. *-commutative 47.3%

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

       5. unpow-prod-down 59.6%

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

       6. pow1/2 59.6%

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

       7. pow1/2 59.6%

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

   13. Applied egg-rr 59.6%

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

4. Final simplification 27.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.6 \cdot 10^{-36}:\\ \;\;\;\;\frac{\sqrt{\left(\left(-4 \cdot A\right) \cdot \left(F \cdot C\right)\right) \cdot \left(4 \cdot C\right)}}{4 \cdot \left(C \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(-\sqrt{\frac{2}{B}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 41.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 1.6 \cdot 10^{-36}:\\ \;\;\;\;\frac{\sqrt{\left(\left(-4 \cdot A\right) \cdot \left(F \cdot C\right)\right) \cdot \left(4 \cdot C\right)}}{4 \cdot \left(C \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\left|F \cdot \frac{2}{B\_m}\right|}\\ \end{array} \end{array} \]

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 1.6e-36)
   (/ (sqrt (* (* (* -4.0 A) (* F C)) (* 4.0 C))) (* 4.0 (* C A)))
   (- (sqrt (fabs (* 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 <= 1.6e-36) {
		tmp = sqrt((((-4.0 * A) * (F * C)) * (4.0 * C))) / (4.0 * (C * A));
	} else {
		tmp = -sqrt(fabs((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 <= 1.6d-36) then
        tmp = sqrt(((((-4.0d0) * a) * (f * c)) * (4.0d0 * c))) / (4.0d0 * (c * a))
    else
        tmp = -sqrt(abs((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 <= 1.6e-36) {
		tmp = Math.sqrt((((-4.0 * A) * (F * C)) * (4.0 * C))) / (4.0 * (C * A));
	} else {
		tmp = -Math.sqrt(Math.abs((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 <= 1.6e-36:
		tmp = math.sqrt((((-4.0 * A) * (F * C)) * (4.0 * C))) / (4.0 * (C * A))
	else:
		tmp = -math.sqrt(math.fabs((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 <= 1.6e-36)
		tmp = Float64(sqrt(Float64(Float64(Float64(-4.0 * A) * Float64(F * C)) * Float64(4.0 * C))) / Float64(4.0 * Float64(C * A)));
	else
		tmp = Float64(-sqrt(abs(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 <= 1.6e-36)
		tmp = sqrt((((-4.0 * A) * (F * C)) * (4.0 * C))) / (4.0 * (C * A));
	else
		tmp = -sqrt(abs((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, 1.6e-36], N[(N[Sqrt[N[(N[(N[(-4.0 * A), $MachinePrecision] * N[(F * C), $MachinePrecision]), $MachinePrecision] * N[(4.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[Sqrt[N[Abs[N[(F * N[(2.0 / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if B < 1.60000000000000011e-36

   1. Initial program 18.7%

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

   3. Simplified 24.7%

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

   5. Taylor expanded in A around -inf 12.8%

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

   6. Taylor expanded in B around 0 11.3%

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

   7. Taylor expanded in B around 0 11.9%

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

      1. associate-*r* 12.0%

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

   9. Simplified 12.0%

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

   if 1.60000000000000011e-36 < B

   1. Initial program 13.7%

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

   3. Taylor expanded in B around inf 47.0%

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

      1. mul-1-neg 47.0%

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

   5. Simplified 47.0%

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

      1. neg-sub0 47.0%

         \[\leadsto \color{blue}{0 - \sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

      2. sqrt-unprod 47.3%

         \[\leadsto 0 - \color{blue}{\sqrt{\frac{F}{B} \cdot 2}} \]

   7. Applied egg-rr 47.3%

      \[\leadsto \color{blue}{0 - \sqrt{\frac{F}{B} \cdot 2}} \]
   8. Step-by-step derivation

      1. neg-sub0 47.3%

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

   9. Simplified 47.3%

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

       1. add-sqr-sqrt 47.3%

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

       2. pow1/2 47.3%

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

       3. pow1/2 47.3%

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

       4. pow-prod-down 37.6%

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

       5. pow2 37.6%

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

       6. *-commutative 37.6%

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

   11. Applied egg-rr 37.6%

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

       1. unpow1/2 37.6%

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

       2. unpow2 37.6%

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

       3. rem-sqrt-square 47.5%

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

       4. associate-*r/ 47.5%

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

       5. *-commutative 47.5%

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

       6. associate-/l* 47.4%

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

   13. Simplified 47.4%

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

4. Final simplification 23.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.6 \cdot 10^{-36}:\\ \;\;\;\;\frac{\sqrt{\left(\left(-4 \cdot A\right) \cdot \left(F \cdot C\right)\right) \cdot \left(4 \cdot C\right)}}{4 \cdot \left(C \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\left|F \cdot \frac{2}{B}\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 41.7% accurate, 5.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 3.6 \cdot 10^{-40}:\\ \;\;\;\;\frac{\sqrt{\left(\left(-4 \cdot A\right) \cdot \left(F \cdot C\right)\right) \cdot \left(4 \cdot C\right)}}{4 \cdot \left(C \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;-{\left(\frac{2 \cdot F}{B\_m}\right)}^{0.5}\\ \end{array} \end{array} \]

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 3.6e-40)
   (/ (sqrt (* (* (* -4.0 A) (* F C)) (* 4.0 C))) (* 4.0 (* C 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 <= 3.6e-40) {
		tmp = sqrt((((-4.0 * A) * (F * C)) * (4.0 * C))) / (4.0 * (C * 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 <= 3.6d-40) then
        tmp = sqrt(((((-4.0d0) * a) * (f * c)) * (4.0d0 * c))) / (4.0d0 * (c * 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 <= 3.6e-40) {
		tmp = Math.sqrt((((-4.0 * A) * (F * C)) * (4.0 * C))) / (4.0 * (C * 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 <= 3.6e-40:
		tmp = math.sqrt((((-4.0 * A) * (F * C)) * (4.0 * C))) / (4.0 * (C * 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 <= 3.6e-40)
		tmp = Float64(sqrt(Float64(Float64(Float64(-4.0 * A) * Float64(F * C)) * Float64(4.0 * C))) / Float64(4.0 * Float64(C * 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 <= 3.6e-40)
		tmp = sqrt((((-4.0 * A) * (F * C)) * (4.0 * C))) / (4.0 * (C * 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, 3.6e-40], N[(N[Sqrt[N[(N[(N[(-4.0 * A), $MachinePrecision] * N[(F * C), $MachinePrecision]), $MachinePrecision] * N[(4.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(4.0 * N[(C * A), $MachinePrecision]), $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 < 3.6e-40

   1. Initial program 18.7%

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

   3. Simplified 24.7%

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

   5. Taylor expanded in A around -inf 12.8%

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

   6. Taylor expanded in B around 0 11.3%

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

   7. Taylor expanded in B around 0 11.9%

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

      1. associate-*r* 12.0%

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

   9. Simplified 12.0%

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

   if 3.6e-40 < B

   1. Initial program 13.7%

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

   3. Taylor expanded in B around inf 47.0%

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

      1. mul-1-neg 47.0%

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

   5. Simplified 47.0%

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

      1. sqrt-div 59.3%

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

   7. Applied egg-rr 59.3%

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

      1. associate-*l/ 59.5%

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

      2. pow1/2 59.5%

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

      3. pow1/2 59.5%

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

      4. pow-prod-down 59.6%

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

   9. Applied egg-rr 59.6%

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

       1. unpow1/2 59.6%

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

   11. Simplified 59.6%

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

       1. sqrt-undiv 47.3%

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

       2. associate-*r/ 47.3%

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

       3. pow1/2 47.3%

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

       4. associate-*r/ 47.3%

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

   13. Applied egg-rr 47.3%

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

4. Final simplification 23.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.6 \cdot 10^{-40}:\\ \;\;\;\;\frac{\sqrt{\left(\left(-4 \cdot A\right) \cdot \left(F \cdot C\right)\right) \cdot \left(4 \cdot C\right)}}{4 \cdot \left(C \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;-{\left(\frac{2 \cdot F}{B}\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 27.9% accurate, 5.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;C \leq 2.3 \cdot 10^{+175}:\\ \;\;\;\;-{\left(\frac{2 \cdot F}{B\_m}\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{F \cdot C}}{B\_m}\\ \end{array} \end{array} \]

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 (<= C 2.3e+175)
   (- (pow (/ (* 2.0 F) B_m) 0.5))
   (* -2.0 (/ (sqrt (* F C)) 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 (C <= 2.3e+175) {
		tmp = -pow(((2.0 * F) / B_m), 0.5);
	} else {
		tmp = -2.0 * (sqrt((F * C)) / 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 (c <= 2.3d+175) then
        tmp = -(((2.0d0 * f) / b_m) ** 0.5d0)
    else
        tmp = (-2.0d0) * (sqrt((f * c)) / 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 (C <= 2.3e+175) {
		tmp = -Math.pow(((2.0 * F) / B_m), 0.5);
	} else {
		tmp = -2.0 * (Math.sqrt((F * C)) / 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 C <= 2.3e+175:
		tmp = -math.pow(((2.0 * F) / B_m), 0.5)
	else:
		tmp = -2.0 * (math.sqrt((F * C)) / 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 (C <= 2.3e+175)
		tmp = Float64(-(Float64(Float64(2.0 * F) / B_m) ^ 0.5));
	else
		tmp = Float64(-2.0 * Float64(sqrt(Float64(F * C)) / 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 (C <= 2.3e+175)
		tmp = -(((2.0 * F) / B_m) ^ 0.5);
	else
		tmp = -2.0 * (sqrt((F * C)) / 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[C, 2.3e+175], (-N[Power[N[(N[(2.0 * F), $MachinePrecision] / B$95$m), $MachinePrecision], 0.5], $MachinePrecision]), N[(-2.0 * N[(N[Sqrt[N[(F * C), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < 2.3e175

   1. Initial program 18.7%

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

   3. Taylor expanded in B around inf 19.7%

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

      1. mul-1-neg 19.7%

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

   5. Simplified 19.7%

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

      1. sqrt-div 24.4%

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

   7. Applied egg-rr 24.4%

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

      1. associate-*l/ 24.4%

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

      2. pow1/2 24.4%

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

      3. pow1/2 24.4%

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

      4. pow-prod-down 24.5%

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

   9. Applied egg-rr 24.5%

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

       1. unpow1/2 24.5%

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

   11. Simplified 24.5%

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

       1. sqrt-undiv 19.9%

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

       2. associate-*r/ 19.9%

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

       3. pow1/2 20.1%

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

       4. associate-*r/ 20.1%

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

   13. Applied egg-rr 20.1%

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

   if 2.3e175 < C

   1. Initial program 1.2%

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

   3. Simplified 17.5%

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

   5. Taylor expanded in A around -inf 17.5%

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

      1. pow1/2 17.5%

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

      2. associate-*r* 17.5%

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

      3. unpow-prod-down 25.1%

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

      4. *-commutative 25.1%

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

      5. fma-undefine 25.1%

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

      6. unpow2 25.1%

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

      7. associate-*r* 25.1%

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

      8. *-commutative 25.1%

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

      9. +-commutative 25.1%

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

      10. *-commutative 25.1%

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

      11. associate-*r* 25.1%

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

      12. fma-define 25.1%

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

      13. pow1/2 25.1%

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

   7. Applied egg-rr 25.1%

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

      1. unpow1/2 25.1%

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

   9. Simplified 25.1%

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

   10. Taylor expanded in A around 0 14.5%

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

       1. associate-*l/ 14.6%

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

       2. *-lft-identity 14.6%

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

       3. *-commutative 14.6%

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

   12. Simplified 14.6%

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

4. Final simplification 19.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 2.3 \cdot 10^{+175}:\\ \;\;\;\;-{\left(\frac{2 \cdot F}{B}\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{F \cdot C}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 27.0% accurate, 5.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ -{\left(\frac{2 \cdot F}{B\_m}\right)}^{0.5} \end{array} \]

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 (- (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) {
	return -pow(((2.0 * F) / B_m), 0.5);
}

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 = -(((2.0d0 * f) / b_m) ** 0.5d0)
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.pow(((2.0 * F) / B_m), 0.5);
}

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.pow(((2.0 * F) / B_m), 0.5)

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(-(Float64(Float64(2.0 * F) / B_m) ^ 0.5))
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 = -(((2.0 * F) / B_m) ^ 0.5);
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[Power[N[(N[(2.0 * F), $MachinePrecision] / B$95$m), $MachinePrecision], 0.5], $MachinePrecision])

Derivation

1. Initial program 17.1%

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

3. Taylor expanded in B around inf 18.2%

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

   1. mul-1-neg 18.2%

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

5. Simplified 18.2%

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

   1. sqrt-div 22.3%

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

7. Applied egg-rr 22.3%

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

   1. associate-*l/ 22.4%

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

   2. pow1/2 22.4%

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

   3. pow1/2 22.4%

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

   4. pow-prod-down 22.4%

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

9. Applied egg-rr 22.4%

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

    1. unpow1/2 22.4%

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

11. Simplified 22.4%

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

    1. sqrt-undiv 18.3%

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

    2. associate-*r/ 18.3%

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

    3. pow1/2 18.5%

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

    4. associate-*r/ 18.5%

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

13. Applied egg-rr 18.5%

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

14. Final simplification 18.5%

    \[\leadsto -{\left(\frac{2 \cdot F}{B}\right)}^{0.5} \]
15. Add Preprocessing

Alternative 11: 27.0% accurate, 5.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ -{\left(2 \cdot \frac{F}{B\_m}\right)}^{0.5} \end{array} \]

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 (- (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) {
	return -pow((2.0 * (F / B_m)), 0.5);
}

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 = -((2.0d0 * (f / b_m)) ** 0.5d0)
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.pow((2.0 * (F / B_m)), 0.5);
}

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.pow((2.0 * (F / B_m)), 0.5)

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(-(Float64(2.0 * Float64(F / B_m)) ^ 0.5))
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 = -((2.0 * (F / B_m)) ^ 0.5);
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[Power[N[(2.0 * N[(F / B$95$m), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision])

Derivation

1. Initial program 17.1%

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

3. Taylor expanded in B around inf 18.2%

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

   1. mul-1-neg 18.2%

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

5. Simplified 18.2%

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

   1. sqrt-unprod 18.3%

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

   2. pow1/2 18.5%

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

7. Applied egg-rr 18.5%

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

8. Final simplification 18.5%

   \[\leadsto -{\left(2 \cdot \frac{F}{B}\right)}^{0.5} \]
9. Add Preprocessing

Alternative 12: 27.0% accurate, 6.0× speedup

Math

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

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 (* 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) {
	return -sqrt((2.0 * (F / B_m)));
}

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((2.0d0 * (f / b_m)))
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((2.0 * (F / B_m)));
}

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((2.0 * (F / B_m)))

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(2.0 * Float64(F / B_m))))
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((2.0 * (F / B_m)));
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[(2.0 * N[(F / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])

Derivation

1. Initial program 17.1%

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

3. Taylor expanded in B around inf 18.2%

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

   1. mul-1-neg 18.2%

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

5. Simplified 18.2%

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

   1. neg-sub0 18.2%

      \[\leadsto \color{blue}{0 - \sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

   2. sqrt-unprod 18.3%

      \[\leadsto 0 - \color{blue}{\sqrt{\frac{F}{B} \cdot 2}} \]

7. Applied egg-rr 18.3%

   \[\leadsto \color{blue}{0 - \sqrt{\frac{F}{B} \cdot 2}} \]
8. Step-by-step derivation

   1. neg-sub0 18.3%

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

9. Simplified 18.3%

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

10. Final simplification 18.3%

    \[\leadsto -\sqrt{2 \cdot \frac{F}{B}} \]
11. Add Preprocessing

Alternative 13: 27.0% accurate, 6.0× speedup

Math

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

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 (/ 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) {
	return -sqrt((F * (2.0 / B_m)));
}

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 * (2.0d0 / b_m)))
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 * (2.0 / B_m)));
}

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 * (2.0 / B_m)))

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(F * Float64(2.0 / B_m))))
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 * (2.0 / B_m)));
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 * N[(2.0 / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])

Derivation

1. Initial program 17.1%

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

3. Taylor expanded in B around inf 18.2%

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

   1. mul-1-neg 18.2%

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

5. Simplified 18.2%

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

   1. neg-sub0 18.2%

      \[\leadsto \color{blue}{0 - \sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

   2. sqrt-unprod 18.3%

      \[\leadsto 0 - \color{blue}{\sqrt{\frac{F}{B} \cdot 2}} \]

7. Applied egg-rr 18.3%

   \[\leadsto \color{blue}{0 - \sqrt{\frac{F}{B} \cdot 2}} \]
8. Step-by-step derivation

   1. neg-sub0 18.3%

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

9. Simplified 18.3%

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

10. Taylor expanded in F around 0 18.3%

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

    1. associate-*r/ 18.3%

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

    2. *-commutative 18.3%

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

    3. associate-/l* 18.3%

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

12. Simplified 18.3%

    \[\leadsto -\sqrt{\color{blue}{F \cdot \frac{2}{B}}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024129 
(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))))