ABCF->ab-angle a

Percentage Accurate: 19.8% → 52.0%

Time: 24.1s

Alternatives: 19

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: 19.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 52.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := -t\_0\\ t_2 := F \cdot t\_0\\ t_3 := \sqrt{2 \cdot t\_2}\\ \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-288}:\\ \;\;\;\;\frac{t\_3 \cdot \left(-\sqrt{-0.5 \cdot \frac{{B\_m}^{2}}{A} + 2 \cdot C}\right)}{t\_0}\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{-128}:\\ \;\;\;\;\frac{\sqrt{A + \left(C + \mathsf{hypot}\left(A - C, B\_m\right)\right)} \cdot t\_3}{t\_1}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+77}:\\ \;\;\;\;\frac{\sqrt{t\_2 \cdot \left(4 \cdot C\right)}}{t\_1}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+275}:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(A + C\right) + \mathsf{hypot}\left(B\_m, A - C\right)}{\mathsf{fma}\left(-4, A \cdot C, {B\_m}^{2}\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot F} \cdot \left(-{B\_m}^{-0.5}\right)\\ \end{array} \end{array} \]

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 (fma B_m B_m (* A (* C -4.0))))
        (t_1 (- t_0))
        (t_2 (* F t_0))
        (t_3 (sqrt (* 2.0 t_2))))
   (if (<= (pow B_m 2.0) 2e-288)
     (/ (* t_3 (- (sqrt (+ (* -0.5 (/ (pow B_m 2.0) A)) (* 2.0 C))))) t_0)
     (if (<= (pow B_m 2.0) 1e-128)
       (/ (* (sqrt (+ A (+ C (hypot (- A C) B_m)))) t_3) t_1)
       (if (<= (pow B_m 2.0) 5e+77)
         (/ (sqrt (* t_2 (* 4.0 C))) t_1)
         (if (<= (pow B_m 2.0) 5e+275)
           (*
            (sqrt
             (*
              F
              (/
               (+ (+ A C) (hypot B_m (- A C)))
               (fma -4.0 (* A C) (pow B_m 2.0)))))
            (- (sqrt 2.0)))
           (* (sqrt (* 2.0 F)) (- (pow 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 t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_1 = -t_0;
	double t_2 = F * t_0;
	double t_3 = sqrt((2.0 * t_2));
	double tmp;
	if (pow(B_m, 2.0) <= 2e-288) {
		tmp = (t_3 * -sqrt(((-0.5 * (pow(B_m, 2.0) / A)) + (2.0 * C)))) / t_0;
	} else if (pow(B_m, 2.0) <= 1e-128) {
		tmp = (sqrt((A + (C + hypot((A - C), B_m)))) * t_3) / t_1;
	} else if (pow(B_m, 2.0) <= 5e+77) {
		tmp = sqrt((t_2 * (4.0 * C))) / t_1;
	} else if (pow(B_m, 2.0) <= 5e+275) {
		tmp = sqrt((F * (((A + C) + hypot(B_m, (A - C))) / fma(-4.0, (A * C), pow(B_m, 2.0))))) * -sqrt(2.0);
	} else {
		tmp = sqrt((2.0 * F)) * -pow(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)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_1 = Float64(-t_0)
	t_2 = Float64(F * t_0)
	t_3 = sqrt(Float64(2.0 * t_2))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-288)
		tmp = Float64(Float64(t_3 * Float64(-sqrt(Float64(Float64(-0.5 * Float64((B_m ^ 2.0) / A)) + Float64(2.0 * C))))) / t_0);
	elseif ((B_m ^ 2.0) <= 1e-128)
		tmp = Float64(Float64(sqrt(Float64(A + Float64(C + hypot(Float64(A - C), B_m)))) * t_3) / t_1);
	elseif ((B_m ^ 2.0) <= 5e+77)
		tmp = Float64(sqrt(Float64(t_2 * Float64(4.0 * C))) / t_1);
	elseif ((B_m ^ 2.0) <= 5e+275)
		tmp = Float64(sqrt(Float64(F * Float64(Float64(Float64(A + C) + hypot(B_m, Float64(A - C))) / fma(-4.0, Float64(A * C), (B_m ^ 2.0))))) * Float64(-sqrt(2.0)));
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) * Float64(-(B_m ^ -0.5)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-t$95$0)}, Block[{t$95$2 = N[(F * t$95$0), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(2.0 * t$95$2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-288], N[(N[(t$95$3 * (-N[Sqrt[N[(N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-128], N[(N[(N[Sqrt[N[(A + N[(C + N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$3), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+77], N[(N[Sqrt[N[(t$95$2 * N[(4.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+275], N[(N[Sqrt[N[(F * N[(N[(N[(A + C), $MachinePrecision] + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] / N[(-4.0 * N[(A * C), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] * (-N[Power[B$95$m, -0.5], $MachinePrecision])), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2.00000000000000012e-288

   1. Initial program 22.1%

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

   3. Simplified 32.6%

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

      1. associate-*r* 32.6%

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

      2. associate-+r+ 30.9%

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

      3. hypot-undefine 22.1%

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

      4. unpow2 22.1%

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

      5. unpow2 22.1%

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

      6. +-commutative 22.1%

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

      7. sqrt-prod 23.5%

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

      8. *-commutative 23.5%

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

      9. associate-+l+ 24.9%

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

   6. Applied egg-rr 37.9%

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

   7. Taylor expanded in A around -inf 27.7%

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

   if 2.00000000000000012e-288 < (pow.f64 B #s(literal 2 binary64)) < 1.00000000000000005e-128

   1. Initial program 36.6%

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

   3. Simplified 45.1%

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

      1. associate-*r* 45.1%

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

      2. associate-+r+ 45.0%

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

      3. hypot-undefine 36.6%

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

      4. unpow2 36.6%

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

      5. unpow2 36.6%

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

      6. +-commutative 36.6%

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

      7. sqrt-prod 41.9%

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

      8. *-commutative 41.9%

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

      9. associate-+l+ 41.9%

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

   6. Applied egg-rr 58.8%

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

   if 1.00000000000000005e-128 < (pow.f64 B #s(literal 2 binary64)) < 5.00000000000000004e77

   1. Initial program 26.3%

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

   3. Simplified 38.0%

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

   5. Taylor expanded in A around -inf 37.0%

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

      1. *-commutative 37.0%

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

   7. Simplified 37.0%

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

   if 5.00000000000000004e77 < (pow.f64 B #s(literal 2 binary64)) < 5.0000000000000003e275

   1. Initial program 18.9%

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

   3. Taylor expanded in F around 0 24.7%

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

   5. Simplified 56.0%

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

   if 5.0000000000000003e275 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 3.0%

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

   3. Taylor expanded in B around inf 26.4%

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

      1. mul-1-neg 26.4%

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

      2. *-commutative 26.4%

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

   5. Simplified 26.4%

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

      1. pow1/2 26.4%

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

      2. div-inv 26.4%

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

      3. unpow-prod-down 39.2%

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

      4. pow1/2 39.2%

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

   7. Applied egg-rr 39.2%

      \[\leadsto -\sqrt{2} \cdot \color{blue}{\left(\sqrt{F} \cdot {\left(\frac{1}{B}\right)}^{0.5}\right)} \]
   8. Step-by-step derivation

      1. unpow1/2 39.2%

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

   9. Simplified 39.2%

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

       1. pow1/2 39.2%

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

       2. exp-to-pow 39.3%

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

       3. associate-*r* 39.2%

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

       4. distribute-rgt-neg-in 39.2%

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

       5. exp-to-pow 39.1%

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

       6. pow1/2 39.1%

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

       7. sqrt-unprod 39.4%

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

       8. inv-pow 39.4%

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

       9. sqrt-pow1 39.4%

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

       10. metadata-eval 39.4%

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

   11. Applied egg-rr 39.4%

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

       1. distribute-rgt-neg-out 39.4%

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

       2. distribute-lft-neg-out 39.4%

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

       3. *-commutative 39.4%

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

       4. *-commutative 39.4%

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

   13. Simplified 39.4%

       \[\leadsto \color{blue}{{B}^{-0.5} \cdot \left(-\sqrt{F \cdot 2}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 40.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-288}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)} \cdot \left(-\sqrt{-0.5 \cdot \frac{{B}^{2}}{A} + 2 \cdot C}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 10^{-128}:\\ \;\;\;\;\frac{\sqrt{A + \left(C + \mathsf{hypot}\left(A - C, B\right)\right)} \cdot \sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+77}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(4 \cdot C\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+275}:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot F} \cdot \left(-{B}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 58.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_1\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_1 - {B\_m}^{2}}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-229}:\\ \;\;\;\;\frac{\left(\sqrt{F} \cdot \sqrt{2 \cdot t\_0}\right) \cdot \sqrt{A + \left(C + \mathsf{hypot}\left(A - C, B\_m\right)\right)}}{-t\_0}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot t\_0\right)} \cdot \left(-\sqrt{-0.5 \cdot \frac{{B\_m}^{2}}{A} + 2 \cdot C}\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{B\_m}} \cdot \left(-\sqrt{2 \cdot F}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0))))
        (t_1 (* (* 4.0 A) C))
        (t_2
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_1) F))
            (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_1 (pow B_m 2.0)))))
   (if (<= t_2 -5e-229)
     (/
      (*
       (* (sqrt F) (sqrt (* 2.0 t_0)))
       (sqrt (+ A (+ C (hypot (- A C) B_m)))))
      (- t_0))
     (if (<= t_2 INFINITY)
       (/
        (*
         (sqrt (* 2.0 (* F t_0)))
         (- (sqrt (+ (* -0.5 (/ (pow B_m 2.0) A)) (* 2.0 C)))))
        t_0)
       (* (sqrt (/ 1.0 B_m)) (- (sqrt (* 2.0 F))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_1 = (4.0 * A) * C;
	double t_2 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_1) * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_1 - pow(B_m, 2.0));
	double tmp;
	if (t_2 <= -5e-229) {
		tmp = ((sqrt(F) * sqrt((2.0 * t_0))) * sqrt((A + (C + hypot((A - C), B_m))))) / -t_0;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = (sqrt((2.0 * (F * t_0))) * -sqrt(((-0.5 * (pow(B_m, 2.0) / A)) + (2.0 * C)))) / t_0;
	} else {
		tmp = sqrt((1.0 / B_m)) * -sqrt((2.0 * F));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_1 = Float64(Float64(4.0 * A) * C)
	t_2 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_1) * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_1 - (B_m ^ 2.0)))
	tmp = 0.0
	if (t_2 <= -5e-229)
		tmp = Float64(Float64(Float64(sqrt(F) * sqrt(Float64(2.0 * t_0))) * sqrt(Float64(A + Float64(C + hypot(Float64(A - C), B_m))))) / Float64(-t_0));
	elseif (t_2 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(F * t_0))) * Float64(-sqrt(Float64(Float64(-0.5 * Float64((B_m ^ 2.0) / A)) + Float64(2.0 * C))))) / t_0);
	else
		tmp = Float64(sqrt(Float64(1.0 / B_m)) * Float64(-sqrt(Float64(2.0 * F))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$1 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e-229], N[(N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(A + N[(C + N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[(N[Sqrt[N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[Sqrt[N[(1.0 / B$95$m), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -5.00000000000000016e-229

   1. Initial program 42.5%

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

   3. Simplified 55.0%

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

      1. associate-*r* 55.0%

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

      2. associate-+r+ 54.5%

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

      3. hypot-undefine 42.5%

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

      4. unpow2 42.5%

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

      5. unpow2 42.5%

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

      6. +-commutative 42.5%

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

      7. sqrt-prod 49.5%

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

      8. *-commutative 49.5%

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

      9. associate-+l+ 49.5%

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

   6. Applied egg-rr 69.0%

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

      1. pow1/2 69.0%

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

      2. associate-*l* 69.0%

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

      3. unpow-prod-down 80.0%

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

      4. pow1/2 80.0%

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

   8. Applied egg-rr 80.0%

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

      1. unpow1/2 80.0%

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

   10. Simplified 80.0%

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

   if -5.00000000000000016e-229 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 23.8%

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

   3. Simplified 33.2%

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

      1. associate-*r* 33.2%

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

      2. associate-+r+ 31.7%

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

      3. hypot-undefine 23.8%

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

      4. unpow2 23.8%

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

      5. unpow2 23.8%

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

      6. +-commutative 23.8%

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

      7. sqrt-prod 25.3%

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

      8. *-commutative 25.3%

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

      9. associate-+l+ 27.3%

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

   6. Applied egg-rr 40.0%

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

   7. Taylor expanded in A around -inf 35.3%

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

   if +inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)))

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf 16.8%

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

      1. mul-1-neg 16.8%

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

      2. *-commutative 16.8%

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

   5. Simplified 16.8%

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

      1. pow1/2 16.9%

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

      2. div-inv 16.9%

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

      3. unpow-prod-down 23.7%

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

      4. pow1/2 23.7%

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

   7. Applied egg-rr 23.7%

      \[\leadsto -\sqrt{2} \cdot \color{blue}{\left(\sqrt{F} \cdot {\left(\frac{1}{B}\right)}^{0.5}\right)} \]
   8. Step-by-step derivation

      1. unpow1/2 23.7%

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

   9. Simplified 23.7%

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

       1. pow1/2 23.7%

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

       2. exp-to-pow 23.7%

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

       3. associate-*r* 23.7%

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

       4. distribute-rgt-neg-in 23.7%

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

       5. exp-to-pow 23.7%

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

       6. pow1/2 23.7%

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

       7. sqrt-unprod 23.8%

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

       8. inv-pow 23.8%

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

       9. sqrt-pow1 23.8%

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

       10. metadata-eval 23.8%

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

   11. Applied egg-rr 23.8%

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

       1. distribute-rgt-neg-out 23.8%

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

       2. distribute-lft-neg-out 23.8%

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

       3. *-commutative 23.8%

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

       4. *-commutative 23.8%

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

   13. Simplified 23.8%

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

   14. Taylor expanded in B around 0 23.8%

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

4. Final simplification 44.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -5 \cdot 10^{-229}:\\ \;\;\;\;\frac{\left(\sqrt{F} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \cdot \sqrt{A + \left(C + \mathsf{hypot}\left(A - C, B\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)} \cdot \left(-\sqrt{-0.5 \cdot \frac{{B}^{2}}{A} + 2 \cdot C}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{B}} \cdot \left(-\sqrt{2 \cdot F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 52.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := F \cdot t\_0\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{-128}:\\ \;\;\;\;\frac{\sqrt{2 \cdot t\_1} \cdot \left(-\sqrt{2 \cdot C}\right)}{t\_0}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+77}:\\ \;\;\;\;\frac{\sqrt{t\_1 \cdot \left(4 \cdot C\right)}}{-t\_0}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+275}:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(A + C\right) + \mathsf{hypot}\left(B\_m, A - C\right)}{\mathsf{fma}\left(-4, A \cdot C, {B\_m}^{2}\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot F} \cdot \left(-{B\_m}^{-0.5}\right)\\ \end{array} \end{array} \]

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 (fma B_m B_m (* A (* C -4.0)))) (t_1 (* F t_0)))
   (if (<= (pow B_m 2.0) 1e-128)
     (/ (* (sqrt (* 2.0 t_1)) (- (sqrt (* 2.0 C)))) t_0)
     (if (<= (pow B_m 2.0) 5e+77)
       (/ (sqrt (* t_1 (* 4.0 C))) (- t_0))
       (if (<= (pow B_m 2.0) 5e+275)
         (*
          (sqrt
           (*
            F
            (/
             (+ (+ A C) (hypot B_m (- A C)))
             (fma -4.0 (* A C) (pow B_m 2.0)))))
          (- (sqrt 2.0)))
         (* (sqrt (* 2.0 F)) (- (pow 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 t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_1 = F * t_0;
	double tmp;
	if (pow(B_m, 2.0) <= 1e-128) {
		tmp = (sqrt((2.0 * t_1)) * -sqrt((2.0 * C))) / t_0;
	} else if (pow(B_m, 2.0) <= 5e+77) {
		tmp = sqrt((t_1 * (4.0 * C))) / -t_0;
	} else if (pow(B_m, 2.0) <= 5e+275) {
		tmp = sqrt((F * (((A + C) + hypot(B_m, (A - C))) / fma(-4.0, (A * C), pow(B_m, 2.0))))) * -sqrt(2.0);
	} else {
		tmp = sqrt((2.0 * F)) * -pow(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)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_1 = Float64(F * t_0)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-128)
		tmp = Float64(Float64(sqrt(Float64(2.0 * t_1)) * Float64(-sqrt(Float64(2.0 * C)))) / t_0);
	elseif ((B_m ^ 2.0) <= 5e+77)
		tmp = Float64(sqrt(Float64(t_1 * Float64(4.0 * C))) / Float64(-t_0));
	elseif ((B_m ^ 2.0) <= 5e+275)
		tmp = Float64(sqrt(Float64(F * Float64(Float64(Float64(A + C) + hypot(B_m, Float64(A - C))) / fma(-4.0, Float64(A * C), (B_m ^ 2.0))))) * Float64(-sqrt(2.0)));
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) * Float64(-(B_m ^ -0.5)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(F * t$95$0), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-128], N[(N[(N[Sqrt[N[(2.0 * t$95$1), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+77], N[(N[Sqrt[N[(t$95$1 * N[(4.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+275], N[(N[Sqrt[N[(F * N[(N[(N[(A + C), $MachinePrecision] + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] / N[(-4.0 * N[(A * C), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] * (-N[Power[B$95$m, -0.5], $MachinePrecision])), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.00000000000000005e-128

   1. Initial program 26.5%

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

   3. Simplified 36.4%

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

      1. associate-*r* 36.4%

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

      2. associate-+r+ 35.3%

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

      3. hypot-undefine 26.5%

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

      4. unpow2 26.5%

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

      5. unpow2 26.5%

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

      6. +-commutative 26.5%

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

      7. sqrt-prod 29.1%

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

      8. *-commutative 29.1%

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

      9. associate-+l+ 30.1%

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

   6. Applied egg-rr 44.3%

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

   7. Taylor expanded in A around -inf 26.0%

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

   if 1.00000000000000005e-128 < (pow.f64 B #s(literal 2 binary64)) < 5.00000000000000004e77

   1. Initial program 26.3%

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

   3. Simplified 38.0%

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

   5. Taylor expanded in A around -inf 37.0%

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

      1. *-commutative 37.0%

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

   7. Simplified 37.0%

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

   if 5.00000000000000004e77 < (pow.f64 B #s(literal 2 binary64)) < 5.0000000000000003e275

   1. Initial program 18.9%

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

   3. Taylor expanded in F around 0 24.7%

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

   5. Simplified 56.0%

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

   if 5.0000000000000003e275 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 3.0%

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

   3. Taylor expanded in B around inf 26.4%

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

      1. mul-1-neg 26.4%

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

      2. *-commutative 26.4%

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

   5. Simplified 26.4%

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

      1. pow1/2 26.4%

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

      2. div-inv 26.4%

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

      3. unpow-prod-down 39.2%

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

      4. pow1/2 39.2%

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

   7. Applied egg-rr 39.2%

      \[\leadsto -\sqrt{2} \cdot \color{blue}{\left(\sqrt{F} \cdot {\left(\frac{1}{B}\right)}^{0.5}\right)} \]
   8. Step-by-step derivation

      1. unpow1/2 39.2%

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

   9. Simplified 39.2%

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

       1. pow1/2 39.2%

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

       2. exp-to-pow 39.3%

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

       3. associate-*r* 39.2%

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

       4. distribute-rgt-neg-in 39.2%

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

       5. exp-to-pow 39.1%

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

       6. pow1/2 39.1%

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

       7. sqrt-unprod 39.4%

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

       8. inv-pow 39.4%

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

       9. sqrt-pow1 39.4%

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

       10. metadata-eval 39.4%

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

   11. Applied egg-rr 39.4%

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

       1. distribute-rgt-neg-out 39.4%

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

       2. distribute-lft-neg-out 39.4%

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

       3. *-commutative 39.4%

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

       4. *-commutative 39.4%

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

   13. Simplified 39.4%

       \[\leadsto \color{blue}{{B}^{-0.5} \cdot \left(-\sqrt{F \cdot 2}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 35.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-128}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)} \cdot \left(-\sqrt{2 \cdot C}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+77}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(4 \cdot C\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+275}:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot F} \cdot \left(-{B}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 52.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := -t\_0\\ t_2 := F \cdot t\_0\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{-128}:\\ \;\;\;\;\frac{\sqrt{2 \cdot t\_2} \cdot \left(-\sqrt{2 \cdot C}\right)}{t\_0}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+77}:\\ \;\;\;\;\frac{\sqrt{t\_2 \cdot \left(4 \cdot C\right)}}{t\_1}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+275}:\\ \;\;\;\;\frac{\left(B\_m \cdot \sqrt{F}\right) \cdot \sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(A - C, B\_m\right)\right)\right)}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot F} \cdot \left(-{B\_m}^{-0.5}\right)\\ \end{array} \end{array} \]

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 (fma B_m B_m (* A (* C -4.0)))) (t_1 (- t_0)) (t_2 (* F t_0)))
   (if (<= (pow B_m 2.0) 1e-128)
     (/ (* (sqrt (* 2.0 t_2)) (- (sqrt (* 2.0 C)))) t_0)
     (if (<= (pow B_m 2.0) 5e+77)
       (/ (sqrt (* t_2 (* 4.0 C))) t_1)
       (if (<= (pow B_m 2.0) 5e+275)
         (/
          (* (* B_m (sqrt F)) (sqrt (* 2.0 (+ A (+ C (hypot (- A C) B_m))))))
          t_1)
         (* (sqrt (* 2.0 F)) (- (pow 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 t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_1 = -t_0;
	double t_2 = F * t_0;
	double tmp;
	if (pow(B_m, 2.0) <= 1e-128) {
		tmp = (sqrt((2.0 * t_2)) * -sqrt((2.0 * C))) / t_0;
	} else if (pow(B_m, 2.0) <= 5e+77) {
		tmp = sqrt((t_2 * (4.0 * C))) / t_1;
	} else if (pow(B_m, 2.0) <= 5e+275) {
		tmp = ((B_m * sqrt(F)) * sqrt((2.0 * (A + (C + hypot((A - C), B_m)))))) / t_1;
	} else {
		tmp = sqrt((2.0 * F)) * -pow(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)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_1 = Float64(-t_0)
	t_2 = Float64(F * t_0)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-128)
		tmp = Float64(Float64(sqrt(Float64(2.0 * t_2)) * Float64(-sqrt(Float64(2.0 * C)))) / t_0);
	elseif ((B_m ^ 2.0) <= 5e+77)
		tmp = Float64(sqrt(Float64(t_2 * Float64(4.0 * C))) / t_1);
	elseif ((B_m ^ 2.0) <= 5e+275)
		tmp = Float64(Float64(Float64(B_m * sqrt(F)) * sqrt(Float64(2.0 * Float64(A + Float64(C + hypot(Float64(A - C), B_m)))))) / t_1);
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) * Float64(-(B_m ^ -0.5)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-t$95$0)}, Block[{t$95$2 = N[(F * t$95$0), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-128], N[(N[(N[Sqrt[N[(2.0 * t$95$2), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+77], N[(N[Sqrt[N[(t$95$2 * N[(4.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+275], N[(N[(N[(B$95$m * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(2.0 * N[(A + N[(C + N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] * (-N[Power[B$95$m, -0.5], $MachinePrecision])), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.00000000000000005e-128

   1. Initial program 26.5%

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

   3. Simplified 36.4%

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

      1. associate-*r* 36.4%

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

      2. associate-+r+ 35.3%

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

      3. hypot-undefine 26.5%

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

      4. unpow2 26.5%

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

      5. unpow2 26.5%

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

      6. +-commutative 26.5%

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

      7. sqrt-prod 29.1%

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

      8. *-commutative 29.1%

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

      9. associate-+l+ 30.1%

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

   6. Applied egg-rr 44.3%

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

   7. Taylor expanded in A around -inf 26.0%

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

   if 1.00000000000000005e-128 < (pow.f64 B #s(literal 2 binary64)) < 5.00000000000000004e77

   1. Initial program 26.3%

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

   3. Simplified 38.0%

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

   5. Taylor expanded in A around -inf 37.0%

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

      1. *-commutative 37.0%

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

   7. Simplified 37.0%

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

   if 5.00000000000000004e77 < (pow.f64 B #s(literal 2 binary64)) < 5.0000000000000003e275

   1. Initial program 18.9%

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

   3. Simplified 22.1%

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

      1. sqrt-prod 38.3%

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

      2. sqrt-prod 54.8%

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

      3. fma-undefine 54.8%

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

      4. add-sqr-sqrt 40.4%

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

      5. hypot-define 40.4%

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

      6. hypot-undefine 32.1%

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

      7. unpow2 32.1%

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

      8. unpow2 32.1%

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

      9. +-commutative 32.1%

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

      10. unpow2 32.1%

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

      11. unpow2 32.1%

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

      12. hypot-define 40.4%

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

   6. Applied egg-rr 40.4%

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

   7. Taylor expanded in B around inf 30.6%

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

   if 5.0000000000000003e275 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 3.0%

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

   3. Taylor expanded in B around inf 26.4%

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

      1. mul-1-neg 26.4%

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

      2. *-commutative 26.4%

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

   5. Simplified 26.4%

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

      1. pow1/2 26.4%

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

      2. div-inv 26.4%

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

      3. unpow-prod-down 39.2%

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

      4. pow1/2 39.2%

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

   7. Applied egg-rr 39.2%

      \[\leadsto -\sqrt{2} \cdot \color{blue}{\left(\sqrt{F} \cdot {\left(\frac{1}{B}\right)}^{0.5}\right)} \]
   8. Step-by-step derivation

      1. unpow1/2 39.2%

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

   9. Simplified 39.2%

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

       1. pow1/2 39.2%

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

       2. exp-to-pow 39.3%

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

       3. associate-*r* 39.2%

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

       4. distribute-rgt-neg-in 39.2%

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

       5. exp-to-pow 39.1%

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

       6. pow1/2 39.1%

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

       7. sqrt-unprod 39.4%

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

       8. inv-pow 39.4%

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

       9. sqrt-pow1 39.4%

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

       10. metadata-eval 39.4%

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

   11. Applied egg-rr 39.4%

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

       1. distribute-rgt-neg-out 39.4%

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

       2. distribute-lft-neg-out 39.4%

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

       3. *-commutative 39.4%

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

       4. *-commutative 39.4%

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

   13. Simplified 39.4%

       \[\leadsto \color{blue}{{B}^{-0.5} \cdot \left(-\sqrt{F \cdot 2}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 32.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-128}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)} \cdot \left(-\sqrt{2 \cdot C}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+77}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(4 \cdot C\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+275}:\\ \;\;\;\;\frac{\left(B \cdot \sqrt{F}\right) \cdot \sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(A - C, B\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot F} \cdot \left(-{B}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 53.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := -t\_0\\ \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{+77}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot t\_0\right) \cdot \left(4 \cdot C\right)}}{t\_1}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+275}:\\ \;\;\;\;\frac{\left(B\_m \cdot \sqrt{F}\right) \cdot \sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(A - C, B\_m\right)\right)\right)}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot F} \cdot \left(-{B\_m}^{-0.5}\right)\\ \end{array} \end{array} \]

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 (fma B_m B_m (* A (* C -4.0)))) (t_1 (- t_0)))
   (if (<= (pow B_m 2.0) 5e+77)
     (/ (sqrt (* (* F t_0) (* 4.0 C))) t_1)
     (if (<= (pow B_m 2.0) 5e+275)
       (/
        (* (* B_m (sqrt F)) (sqrt (* 2.0 (+ A (+ C (hypot (- A C) B_m))))))
        t_1)
       (* (sqrt (* 2.0 F)) (- (pow 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 t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_1 = -t_0;
	double tmp;
	if (pow(B_m, 2.0) <= 5e+77) {
		tmp = sqrt(((F * t_0) * (4.0 * C))) / t_1;
	} else if (pow(B_m, 2.0) <= 5e+275) {
		tmp = ((B_m * sqrt(F)) * sqrt((2.0 * (A + (C + hypot((A - C), B_m)))))) / t_1;
	} else {
		tmp = sqrt((2.0 * F)) * -pow(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)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_1 = Float64(-t_0)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e+77)
		tmp = Float64(sqrt(Float64(Float64(F * t_0) * Float64(4.0 * C))) / t_1);
	elseif ((B_m ^ 2.0) <= 5e+275)
		tmp = Float64(Float64(Float64(B_m * sqrt(F)) * sqrt(Float64(2.0 * Float64(A + Float64(C + hypot(Float64(A - C), B_m)))))) / t_1);
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) * Float64(-(B_m ^ -0.5)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-t$95$0)}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+77], N[(N[Sqrt[N[(N[(F * t$95$0), $MachinePrecision] * N[(4.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+275], N[(N[(N[(B$95$m * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(2.0 * N[(A + N[(C + N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] * (-N[Power[B$95$m, -0.5], $MachinePrecision])), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 5.00000000000000004e77

   1. Initial program 26.5%

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

   3. Simplified 36.8%

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

   5. Taylor expanded in A around -inf 25.7%

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

      1. *-commutative 25.7%

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

   7. Simplified 25.7%

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

   if 5.00000000000000004e77 < (pow.f64 B #s(literal 2 binary64)) < 5.0000000000000003e275

   1. Initial program 18.9%

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

   3. Simplified 22.1%

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

      1. sqrt-prod 38.3%

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

      2. sqrt-prod 54.8%

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

      3. fma-undefine 54.8%

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

      4. add-sqr-sqrt 40.4%

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

      5. hypot-define 40.4%

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

      6. hypot-undefine 32.1%

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

      7. unpow2 32.1%

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

      8. unpow2 32.1%

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

      9. +-commutative 32.1%

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

      10. unpow2 32.1%

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

      11. unpow2 32.1%

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

      12. hypot-define 40.4%

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

   6. Applied egg-rr 40.4%

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

   7. Taylor expanded in B around inf 30.6%

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

   if 5.0000000000000003e275 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 3.0%

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

   3. Taylor expanded in B around inf 26.4%

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

      1. mul-1-neg 26.4%

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

      2. *-commutative 26.4%

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

   5. Simplified 26.4%

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

      1. pow1/2 26.4%

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

      2. div-inv 26.4%

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

      3. unpow-prod-down 39.2%

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

      4. pow1/2 39.2%

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

   7. Applied egg-rr 39.2%

      \[\leadsto -\sqrt{2} \cdot \color{blue}{\left(\sqrt{F} \cdot {\left(\frac{1}{B}\right)}^{0.5}\right)} \]
   8. Step-by-step derivation

      1. unpow1/2 39.2%

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

   9. Simplified 39.2%

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

       1. pow1/2 39.2%

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

       2. exp-to-pow 39.3%

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

       3. associate-*r* 39.2%

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

       4. distribute-rgt-neg-in 39.2%

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

       5. exp-to-pow 39.1%

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

       6. pow1/2 39.1%

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

       7. sqrt-unprod 39.4%

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

       8. inv-pow 39.4%

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

       9. sqrt-pow1 39.4%

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

       10. metadata-eval 39.4%

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

   11. Applied egg-rr 39.4%

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

       1. distribute-rgt-neg-out 39.4%

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

       2. distribute-lft-neg-out 39.4%

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

       3. *-commutative 39.4%

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

       4. *-commutative 39.4%

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

   13. Simplified 39.4%

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

4. Final simplification 30.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{+77}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(4 \cdot C\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+275}:\\ \;\;\;\;\frac{\left(B \cdot \sqrt{F}\right) \cdot \sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(A - C, B\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot F} \cdot \left(-{B}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{+77}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot t\_0\right) \cdot \left(4 \cdot C\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{B\_m}} \cdot \left(-\sqrt{2 \cdot F}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= (pow B_m 2.0) 5e+77)
     (/ (sqrt (* (* F t_0) (* 4.0 C))) (- t_0))
     (* (sqrt (/ 1.0 B_m)) (- (sqrt (* 2.0 F)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (pow(B_m, 2.0) <= 5e+77) {
		tmp = sqrt(((F * t_0) * (4.0 * C))) / -t_0;
	} else {
		tmp = sqrt((1.0 / B_m)) * -sqrt((2.0 * F));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e+77)
		tmp = Float64(sqrt(Float64(Float64(F * t_0) * Float64(4.0 * C))) / Float64(-t_0));
	else
		tmp = Float64(sqrt(Float64(1.0 / B_m)) * Float64(-sqrt(Float64(2.0 * F))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+77], N[(N[Sqrt[N[(N[(F * t$95$0), $MachinePrecision] * N[(4.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[Sqrt[N[(1.0 / B$95$m), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 5.00000000000000004e77

   1. Initial program 26.5%

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

   3. Simplified 36.8%

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

   5. Taylor expanded in A around -inf 25.7%

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

      1. *-commutative 25.7%

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

   7. Simplified 25.7%

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

   if 5.00000000000000004e77 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 8.3%

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

   3. Taylor expanded in B around inf 23.2%

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

      1. mul-1-neg 23.2%

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

      2. *-commutative 23.2%

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

   5. Simplified 23.2%

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

      1. pow1/2 23.2%

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

      2. div-inv 23.2%

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

      3. unpow-prod-down 32.5%

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

      4. pow1/2 32.5%

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

   7. Applied egg-rr 32.5%

      \[\leadsto -\sqrt{2} \cdot \color{blue}{\left(\sqrt{F} \cdot {\left(\frac{1}{B}\right)}^{0.5}\right)} \]
   8. Step-by-step derivation

      1. unpow1/2 32.5%

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

   9. Simplified 32.5%

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

       1. pow1/2 32.5%

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

       2. exp-to-pow 32.5%

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

       3. associate-*r* 32.5%

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

       4. distribute-rgt-neg-in 32.5%

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

       5. exp-to-pow 32.5%

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

       6. pow1/2 32.5%

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

       7. sqrt-unprod 32.7%

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

       8. inv-pow 32.7%

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

       9. sqrt-pow1 32.6%

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

       10. metadata-eval 32.6%

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

   11. Applied egg-rr 32.6%

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

       1. distribute-rgt-neg-out 32.6%

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

       2. distribute-lft-neg-out 32.6%

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

       3. *-commutative 32.6%

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

       4. *-commutative 32.6%

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

   13. Simplified 32.6%

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

   14. Taylor expanded in B around 0 32.7%

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

4. Final simplification 28.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{+77}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(4 \cdot C\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{B}} \cdot \left(-\sqrt{2 \cdot F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 39.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 10^{-176}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A + 2 \cdot C\right)\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B\_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{B\_m}} \cdot \left(-\sqrt{2 \cdot F}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 1e-176)
   (/
    (sqrt (* -8.0 (* A (* C (* F (+ A (* 2.0 C)))))))
    (- (* (* 4.0 A) C) (pow B_m 2.0)))
   (* (sqrt (/ 1.0 B_m)) (- (sqrt (* 2.0 F))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 1e-176) {
		tmp = sqrt((-8.0 * (A * (C * (F * (A + (2.0 * C))))))) / (((4.0 * A) * C) - pow(B_m, 2.0));
	} else {
		tmp = sqrt((1.0 / B_m)) * -sqrt((2.0 * F));
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if ((b_m ** 2.0d0) <= 1d-176) then
        tmp = sqrt(((-8.0d0) * (a * (c * (f * (a + (2.0d0 * c))))))) / (((4.0d0 * a) * c) - (b_m ** 2.0d0))
    else
        tmp = sqrt((1.0d0 / b_m)) * -sqrt((2.0d0 * f))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (Math.pow(B_m, 2.0) <= 1e-176) {
		tmp = Math.sqrt((-8.0 * (A * (C * (F * (A + (2.0 * C))))))) / (((4.0 * A) * C) - Math.pow(B_m, 2.0));
	} else {
		tmp = Math.sqrt((1.0 / B_m)) * -Math.sqrt((2.0 * F));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if math.pow(B_m, 2.0) <= 1e-176:
		tmp = math.sqrt((-8.0 * (A * (C * (F * (A + (2.0 * C))))))) / (((4.0 * A) * C) - math.pow(B_m, 2.0))
	else:
		tmp = math.sqrt((1.0 / B_m)) * -math.sqrt((2.0 * F))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-176)
		tmp = Float64(sqrt(Float64(-8.0 * Float64(A * Float64(C * Float64(F * Float64(A + Float64(2.0 * C))))))) / Float64(Float64(Float64(4.0 * A) * C) - (B_m ^ 2.0)));
	else
		tmp = Float64(sqrt(Float64(1.0 / B_m)) * Float64(-sqrt(Float64(2.0 * F))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 1e-176)
		tmp = sqrt((-8.0 * (A * (C * (F * (A + (2.0 * C))))))) / (((4.0 * A) * C) - (B_m ^ 2.0));
	else
		tmp = sqrt((1.0 / B_m)) * -sqrt((2.0 * F));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-176], N[(N[Sqrt[N[(-8.0 * N[(A * N[(C * N[(F * N[(A + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 / B$95$m), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1e-176

   1. Initial program 26.7%

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

   3. Taylor expanded in C around inf 17.7%

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

   4. Taylor expanded in B around 0 13.5%

      \[\leadsto \frac{-\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A + 2 \cdot C\right)\right)\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   if 1e-176 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 14.3%

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

   3. Taylor expanded in B around inf 18.0%

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

      1. mul-1-neg 18.0%

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

      2. *-commutative 18.0%

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

   5. Simplified 18.0%

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

      1. pow1/2 18.0%

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

      2. div-inv 18.0%

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

      3. unpow-prod-down 24.1%

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

      4. pow1/2 24.1%

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

   7. Applied egg-rr 24.1%

      \[\leadsto -\sqrt{2} \cdot \color{blue}{\left(\sqrt{F} \cdot {\left(\frac{1}{B}\right)}^{0.5}\right)} \]
   8. Step-by-step derivation

      1. unpow1/2 24.1%

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

   9. Simplified 24.1%

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

       1. pow1/2 24.1%

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

       2. exp-to-pow 24.1%

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

       3. associate-*r* 24.1%

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

       4. distribute-rgt-neg-in 24.1%

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

       5. exp-to-pow 24.1%

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

       6. pow1/2 24.1%

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

       7. sqrt-unprod 24.2%

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

       8. inv-pow 24.2%

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

       9. sqrt-pow1 24.2%

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

       10. metadata-eval 24.2%

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

   11. Applied egg-rr 24.2%

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

       1. distribute-rgt-neg-out 24.2%

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

       2. distribute-lft-neg-out 24.2%

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

       3. *-commutative 24.2%

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

       4. *-commutative 24.2%

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

   13. Simplified 24.2%

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

   14. Taylor expanded in B around 0 24.2%

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

4. Final simplification 20.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-176}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A + 2 \cdot C\right)\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{B}} \cdot \left(-\sqrt{2 \cdot F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 43.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;B\_m \leq 9.4 \cdot 10^{-89}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(C + \left(A + C\right)\right)}}{t\_0 - B\_m \cdot B\_m}\\ \mathbf{elif}\;B\_m \leq 7.2 \cdot 10^{+185}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\_m\right)\right)} \cdot \left(-\frac{\sqrt{2}}{B\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot F} \cdot \left(-{B\_m}^{-0.5}\right)\\ \end{array} \end{array} \]

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 (* (* 4.0 A) C)))
   (if (<= B_m 9.4e-89)
     (/
      (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) (+ C (+ A C))))
      (- t_0 (* B_m B_m)))
     (if (<= B_m 7.2e+185)
       (* (sqrt (* F (+ C (hypot C B_m)))) (- (/ (sqrt 2.0) B_m)))
       (* (sqrt (* 2.0 F)) (- (pow 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 t_0 = (4.0 * A) * C;
	double tmp;
	if (B_m <= 9.4e-89) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (C + (A + C)))) / (t_0 - (B_m * B_m));
	} else if (B_m <= 7.2e+185) {
		tmp = sqrt((F * (C + hypot(C, B_m)))) * -(sqrt(2.0) / B_m);
	} else {
		tmp = sqrt((2.0 * F)) * -pow(B_m, -0.5);
	}
	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 = (4.0 * A) * C;
	double tmp;
	if (B_m <= 9.4e-89) {
		tmp = Math.sqrt(((2.0 * ((Math.pow(B_m, 2.0) - t_0) * F)) * (C + (A + C)))) / (t_0 - (B_m * B_m));
	} else if (B_m <= 7.2e+185) {
		tmp = Math.sqrt((F * (C + Math.hypot(C, B_m)))) * -(Math.sqrt(2.0) / B_m);
	} else {
		tmp = Math.sqrt((2.0 * F)) * -Math.pow(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):
	t_0 = (4.0 * A) * C
	tmp = 0
	if B_m <= 9.4e-89:
		tmp = math.sqrt(((2.0 * ((math.pow(B_m, 2.0) - t_0) * F)) * (C + (A + C)))) / (t_0 - (B_m * B_m))
	elif B_m <= 7.2e+185:
		tmp = math.sqrt((F * (C + math.hypot(C, B_m)))) * -(math.sqrt(2.0) / B_m)
	else:
		tmp = math.sqrt((2.0 * F)) * -math.pow(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)
	t_0 = Float64(Float64(4.0 * A) * C)
	tmp = 0.0
	if (B_m <= 9.4e-89)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(C + Float64(A + C)))) / Float64(t_0 - Float64(B_m * B_m)));
	elseif (B_m <= 7.2e+185)
		tmp = Float64(sqrt(Float64(F * Float64(C + hypot(C, B_m)))) * Float64(-Float64(sqrt(2.0) / B_m)));
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) * Float64(-(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)
	t_0 = (4.0 * A) * C;
	tmp = 0.0;
	if (B_m <= 9.4e-89)
		tmp = sqrt(((2.0 * (((B_m ^ 2.0) - t_0) * F)) * (C + (A + C)))) / (t_0 - (B_m * B_m));
	elseif (B_m <= 7.2e+185)
		tmp = sqrt((F * (C + hypot(C, B_m)))) * -(sqrt(2.0) / B_m);
	else
		tmp = sqrt((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_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, If[LessEqual[B$95$m, 9.4e-89], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(C + N[(A + C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 7.2e+185], N[(N[Sqrt[N[(F * N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] * (-N[Power[B$95$m, -0.5], $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 9.39999999999999991e-89

   1. Initial program 20.9%

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

   3. Taylor expanded in C around inf 12.6%

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

      1. unpow2 12.6%

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

   5. Applied egg-rr 12.6%

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

   if 9.39999999999999991e-89 < B < 7.20000000000000058e185

   1. Initial program 21.8%

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

   3. Taylor expanded in A around 0 21.2%

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

      1. mul-1-neg 21.2%

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

      2. *-commutative 21.2%

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

      3. *-commutative 21.2%

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

      4. +-commutative 21.2%

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

      5. unpow2 21.2%

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

      6. unpow2 21.2%

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

      7. hypot-define 27.7%

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

   5. Simplified 27.7%

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

   if 7.20000000000000058e185 < B

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf 58.3%

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

      1. mul-1-neg 58.3%

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

      2. *-commutative 58.3%

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

   5. Simplified 58.3%

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

      1. pow1/2 58.4%

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

      2. div-inv 58.3%

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

      3. unpow-prod-down 88.9%

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

      4. pow1/2 88.9%

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

   7. Applied egg-rr 88.9%

      \[\leadsto -\sqrt{2} \cdot \color{blue}{\left(\sqrt{F} \cdot {\left(\frac{1}{B}\right)}^{0.5}\right)} \]
   8. Step-by-step derivation

      1. unpow1/2 88.9%

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

   9. Simplified 88.9%

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

       1. pow1/2 88.9%

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

       2. exp-to-pow 89.0%

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

       3. associate-*r* 88.9%

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

       4. distribute-rgt-neg-in 88.9%

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

       5. exp-to-pow 88.9%

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

       6. pow1/2 88.9%

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

       7. sqrt-unprod 89.3%

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

       8. inv-pow 89.3%

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

       9. sqrt-pow1 89.3%

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

       10. metadata-eval 89.3%

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

   11. Applied egg-rr 89.3%

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

       1. distribute-rgt-neg-out 89.3%

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

       2. distribute-lft-neg-out 89.3%

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

       3. *-commutative 89.3%

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

       4. *-commutative 89.3%

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

   13. Simplified 89.3%

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

4. Final simplification 23.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 9.4 \cdot 10^{-89}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(C + \left(A + C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;B \leq 7.2 \cdot 10^{+185}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot F} \cdot \left(-{B}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 40.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;B\_m \leq 4 \cdot 10^{+32}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(C + \left(A + C\right)\right)}}{t\_0 - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{B\_m}} \cdot \left(-\sqrt{2 \cdot F}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C)))
   (if (<= B_m 4e+32)
     (/
      (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) (+ C (+ A C))))
      (- t_0 (* B_m B_m)))
     (* (sqrt (/ 1.0 B_m)) (- (sqrt (* 2.0 F)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double tmp;
	if (B_m <= 4e+32) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (C + (A + C)))) / (t_0 - (B_m * B_m));
	} else {
		tmp = sqrt((1.0 / B_m)) * -sqrt((2.0 * F));
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (4.0d0 * a) * c
    if (b_m <= 4d+32) then
        tmp = sqrt(((2.0d0 * (((b_m ** 2.0d0) - t_0) * f)) * (c + (a + c)))) / (t_0 - (b_m * b_m))
    else
        tmp = sqrt((1.0d0 / b_m)) * -sqrt((2.0d0 * f))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double tmp;
	if (B_m <= 4e+32) {
		tmp = Math.sqrt(((2.0 * ((Math.pow(B_m, 2.0) - t_0) * F)) * (C + (A + C)))) / (t_0 - (B_m * B_m));
	} else {
		tmp = Math.sqrt((1.0 / B_m)) * -Math.sqrt((2.0 * F));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = (4.0 * A) * C
	tmp = 0
	if B_m <= 4e+32:
		tmp = math.sqrt(((2.0 * ((math.pow(B_m, 2.0) - t_0) * F)) * (C + (A + C)))) / (t_0 - (B_m * B_m))
	else:
		tmp = math.sqrt((1.0 / B_m)) * -math.sqrt((2.0 * F))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	tmp = 0.0
	if (B_m <= 4e+32)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(C + Float64(A + C)))) / Float64(t_0 - Float64(B_m * B_m)));
	else
		tmp = Float64(sqrt(Float64(1.0 / B_m)) * Float64(-sqrt(Float64(2.0 * F))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = (4.0 * A) * C;
	tmp = 0.0;
	if (B_m <= 4e+32)
		tmp = sqrt(((2.0 * (((B_m ^ 2.0) - t_0) * F)) * (C + (A + C)))) / (t_0 - (B_m * B_m));
	else
		tmp = sqrt((1.0 / B_m)) * -sqrt((2.0 * F));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, If[LessEqual[B$95$m, 4e+32], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(C + N[(A + C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 / B$95$m), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 4.00000000000000021e32

   1. Initial program 21.2%

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

   3. Taylor expanded in C around inf 13.5%

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

      1. unpow2 13.5%

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

   5. Applied egg-rr 13.5%

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

   if 4.00000000000000021e32 < B

   1. Initial program 10.3%

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

   3. Taylor expanded in B around inf 46.3%

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

      1. mul-1-neg 46.3%

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

      2. *-commutative 46.3%

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

   5. Simplified 46.3%

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

      1. pow1/2 46.3%

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

      2. div-inv 46.3%

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

      3. unpow-prod-down 66.9%

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

      4. pow1/2 66.9%

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

   7. Applied egg-rr 66.9%

      \[\leadsto -\sqrt{2} \cdot \color{blue}{\left(\sqrt{F} \cdot {\left(\frac{1}{B}\right)}^{0.5}\right)} \]
   8. Step-by-step derivation

      1. unpow1/2 66.9%

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

   9. Simplified 66.9%

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

       1. pow1/2 66.9%

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

       2. exp-to-pow 67.0%

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

       3. associate-*r* 66.9%

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

       4. distribute-rgt-neg-in 66.9%

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

       5. exp-to-pow 66.9%

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

       6. pow1/2 66.9%

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

       7. sqrt-unprod 67.3%

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

       8. inv-pow 67.3%

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

       9. sqrt-pow1 67.2%

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

       10. metadata-eval 67.2%

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

   11. Applied egg-rr 67.2%

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

       1. distribute-rgt-neg-out 67.2%

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

       2. distribute-lft-neg-out 67.2%

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

       3. *-commutative 67.2%

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

       4. *-commutative 67.2%

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

   13. Simplified 67.2%

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

   14. Taylor expanded in B around 0 67.3%

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

4. Final simplification 24.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 4 \cdot 10^{+32}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(C + \left(A + C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{B}} \cdot \left(-\sqrt{2 \cdot F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 35.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;C \leq 1.3 \cdot 10^{+226}:\\ \;\;\;\;\sqrt{\frac{1}{B\_m}} \cdot \left(-\sqrt{2 \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A + 2 \cdot C\right)} \cdot \left(-\frac{\sqrt{2}}{B\_m}\right)\\ \end{array} \end{array} \]

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 (<= C 1.3e+226)
   (* (sqrt (/ 1.0 B_m)) (- (sqrt (* 2.0 F))))
   (* (sqrt (* F (+ A (* 2.0 C)))) (- (/ (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 (C <= 1.3e+226) {
		tmp = sqrt((1.0 / B_m)) * -sqrt((2.0 * F));
	} else {
		tmp = sqrt((F * (A + (2.0 * C)))) * -(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 (c <= 1.3d+226) then
        tmp = sqrt((1.0d0 / b_m)) * -sqrt((2.0d0 * f))
    else
        tmp = sqrt((f * (a + (2.0d0 * c)))) * -(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 (C <= 1.3e+226) {
		tmp = Math.sqrt((1.0 / B_m)) * -Math.sqrt((2.0 * F));
	} else {
		tmp = Math.sqrt((F * (A + (2.0 * C)))) * -(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 C <= 1.3e+226:
		tmp = math.sqrt((1.0 / B_m)) * -math.sqrt((2.0 * F))
	else:
		tmp = math.sqrt((F * (A + (2.0 * C)))) * -(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 (C <= 1.3e+226)
		tmp = Float64(sqrt(Float64(1.0 / B_m)) * Float64(-sqrt(Float64(2.0 * F))));
	else
		tmp = Float64(sqrt(Float64(F * Float64(A + Float64(2.0 * C)))) * Float64(-Float64(sqrt(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 (C <= 1.3e+226)
		tmp = sqrt((1.0 / B_m)) * -sqrt((2.0 * F));
	else
		tmp = sqrt((F * (A + (2.0 * C)))) * -(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[C, 1.3e+226], N[(N[Sqrt[N[(1.0 / B$95$m), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(F * N[(A + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < 1.3000000000000001e226

   1. Initial program 20.0%

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

   3. Taylor expanded in B around inf 12.9%

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

      1. mul-1-neg 12.9%

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

      2. *-commutative 12.9%

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

   5. Simplified 12.9%

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

      1. pow1/2 13.2%

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

      2. div-inv 13.1%

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

      3. unpow-prod-down 17.3%

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

      4. pow1/2 17.3%

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

   7. Applied egg-rr 17.3%

      \[\leadsto -\sqrt{2} \cdot \color{blue}{\left(\sqrt{F} \cdot {\left(\frac{1}{B}\right)}^{0.5}\right)} \]
   8. Step-by-step derivation

      1. unpow1/2 17.3%

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

   9. Simplified 17.3%

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

       1. pow1/2 17.3%

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

       2. exp-to-pow 17.3%

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

       3. associate-*r* 17.3%

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

       4. distribute-rgt-neg-in 17.3%

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

       5. exp-to-pow 17.3%

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

       6. pow1/2 17.3%

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

       7. sqrt-unprod 17.4%

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

       8. inv-pow 17.4%

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

       9. sqrt-pow1 17.4%

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

       10. metadata-eval 17.4%

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

   11. Applied egg-rr 17.4%

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

       1. distribute-rgt-neg-out 17.4%

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

       2. distribute-lft-neg-out 17.4%

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

       3. *-commutative 17.4%

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

       4. *-commutative 17.4%

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

   13. Simplified 17.4%

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

   14. Taylor expanded in B around 0 17.4%

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

   if 1.3000000000000001e226 < C

   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 C around inf 22.1%

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

   4. Taylor expanded in B around inf 1.6%

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

      1. mul-1-neg 1.6%

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

   6. Simplified 1.6%

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

4. Final simplification 16.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 1.3 \cdot 10^{+226}:\\ \;\;\;\;\sqrt{\frac{1}{B}} \cdot \left(-\sqrt{2 \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A + 2 \cdot C\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 35.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \sqrt{\frac{1}{B\_m}} \cdot \left(-\sqrt{2 \cdot F}\right) \end{array} \]

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 (/ 1.0 B_m)) (- (sqrt (* 2.0 F)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return sqrt((1.0 / B_m)) * -sqrt((2.0 * F));
}

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((1.0d0 / b_m)) * -sqrt((2.0d0 * f))
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((1.0 / B_m)) * -Math.sqrt((2.0 * F));
}

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((1.0 / B_m)) * -math.sqrt((2.0 * F))

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(1.0 / B_m)) * Float64(-sqrt(Float64(2.0 * F))))
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((1.0 / B_m)) * -sqrt((2.0 * F));
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[(N[Sqrt[N[(1.0 / B$95$m), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 19.0%

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

3. Taylor expanded in B around inf 12.3%

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

   1. mul-1-neg 12.3%

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

   2. *-commutative 12.3%

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

5. Simplified 12.3%

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

   1. pow1/2 12.5%

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

   2. div-inv 12.5%

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

   3. unpow-prod-down 16.5%

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

   4. pow1/2 16.5%

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

7. Applied egg-rr 16.5%

   \[\leadsto -\sqrt{2} \cdot \color{blue}{\left(\sqrt{F} \cdot {\left(\frac{1}{B}\right)}^{0.5}\right)} \]
8. Step-by-step derivation

   1. unpow1/2 16.5%

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

9. Simplified 16.5%

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

    1. pow1/2 16.5%

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

    2. exp-to-pow 16.5%

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

    3. associate-*r* 16.5%

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

    4. distribute-rgt-neg-in 16.5%

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

    5. exp-to-pow 16.5%

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

    6. pow1/2 16.5%

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

    7. sqrt-unprod 16.5%

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

    8. inv-pow 16.5%

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

    9. sqrt-pow1 16.5%

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

    10. metadata-eval 16.5%

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

11. Applied egg-rr 16.5%

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

    1. distribute-rgt-neg-out 16.5%

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

    2. distribute-lft-neg-out 16.5%

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

    3. *-commutative 16.5%

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

    4. *-commutative 16.5%

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

13. Simplified 16.5%

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

14. Taylor expanded in B around 0 16.5%

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

15. Final simplification 16.5%

    \[\leadsto \sqrt{\frac{1}{B}} \cdot \left(-\sqrt{2 \cdot F}\right) \]
16. Add Preprocessing

Alternative 12: 35.1% accurate, 3.1× speedup

Math

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

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)) (- (pow 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 sqrt((2.0 * F)) * -pow(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 = sqrt((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.sqrt((2.0 * F)) * -Math.pow(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.sqrt((2.0 * F)) * -math.pow(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(sqrt(Float64(2.0 * F)) * Float64(-(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 = sqrt((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[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] * (-N[Power[B$95$m, -0.5], $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 19.0%

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

3. Taylor expanded in B around inf 12.3%

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

   1. mul-1-neg 12.3%

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

   2. *-commutative 12.3%

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

5. Simplified 12.3%

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

   1. pow1/2 12.5%

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

   2. div-inv 12.5%

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

   3. unpow-prod-down 16.5%

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

   4. pow1/2 16.5%

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

7. Applied egg-rr 16.5%

   \[\leadsto -\sqrt{2} \cdot \color{blue}{\left(\sqrt{F} \cdot {\left(\frac{1}{B}\right)}^{0.5}\right)} \]
8. Step-by-step derivation

   1. unpow1/2 16.5%

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

9. Simplified 16.5%

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

    1. pow1/2 16.5%

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

    2. exp-to-pow 16.5%

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

    3. associate-*r* 16.5%

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

    4. distribute-rgt-neg-in 16.5%

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

    5. exp-to-pow 16.5%

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

    6. pow1/2 16.5%

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

    7. sqrt-unprod 16.5%

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

    8. inv-pow 16.5%

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

    9. sqrt-pow1 16.5%

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

    10. metadata-eval 16.5%

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

11. Applied egg-rr 16.5%

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

    1. distribute-rgt-neg-out 16.5%

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

    2. distribute-lft-neg-out 16.5%

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

    3. *-commutative 16.5%

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

    4. *-commutative 16.5%

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

13. Simplified 16.5%

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

14. Final simplification 16.5%

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

Alternative 13: 27.1% accurate, 3.1× speedup

Math

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

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 (fabs (/ 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 * fabs((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 * abs((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 * Math.abs((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 * math.fabs((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 * abs(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 * abs((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[Abs[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision])

Derivation

1. Initial program 19.0%

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

3. Taylor expanded in B around inf 12.3%

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

   1. mul-1-neg 12.3%

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

   2. *-commutative 12.3%

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

5. Simplified 12.3%

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

   1. *-commutative 12.3%

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

   2. pow1/2 12.5%

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

   3. pow1/2 12.5%

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

   4. pow-prod-down 12.6%

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

7. Applied egg-rr 12.6%

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

   1. unpow1/2 12.3%

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

9. Simplified 12.3%

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

    1. add-sqr-sqrt 12.3%

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

    2. pow1/2 12.3%

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

    3. pow1/2 12.5%

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

    4. pow-prod-down 16.0%

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

    5. pow2 16.0%

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

11. Applied egg-rr 16.0%

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

    1. unpow1/2 16.0%

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

    2. unpow2 16.0%

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

    3. rem-sqrt-square 23.3%

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

13. Simplified 23.3%

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

14. Final simplification 23.3%

    \[\leadsto -\sqrt{2 \cdot \left|\frac{F}{B}\right|} \]
15. Add Preprocessing

Alternative 14: 27.1% accurate, 3.1× speedup

Math

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

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 (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) {
	return -sqrt(fabs((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(abs((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(Math.abs((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(math.fabs((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(abs(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(abs((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[Abs[N[(F * N[(2.0 / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision])

Derivation

1. Initial program 19.0%

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

3. Taylor expanded in B around inf 12.3%

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

   1. mul-1-neg 12.3%

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

   2. *-commutative 12.3%

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

5. Simplified 12.3%

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

   1. *-commutative 12.3%

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

   2. pow1/2 12.5%

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

   3. pow1/2 12.5%

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

   4. pow-prod-down 12.6%

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

7. Applied egg-rr 12.6%

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

   1. unpow1/2 12.3%

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

9. Simplified 12.3%

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

    1. add-sqr-sqrt 12.3%

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

    2. pow1/2 12.3%

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

    3. pow1/2 12.6%

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

    4. pow-prod-down 16.0%

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

    5. pow2 16.0%

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

    6. *-commutative 16.0%

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

    7. clear-num 16.0%

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

    8. un-div-inv 16.0%

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

11. Applied egg-rr 16.0%

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

    1. unpow1/2 16.0%

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

    2. unpow2 16.0%

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

    3. rem-sqrt-square 23.3%

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

    4. associate-/r/ 23.3%

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

13. Simplified 23.3%

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

14. Final simplification 23.3%

    \[\leadsto -\sqrt{\left|F \cdot \frac{2}{B}\right|} \]
15. Add Preprocessing

Alternative 15: 26.9% 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 19.0%

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

3. Taylor expanded in B around inf 12.3%

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

   1. mul-1-neg 12.3%

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

   2. *-commutative 12.3%

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

5. Simplified 12.3%

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

   1. *-commutative 12.3%

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

   2. pow1/2 12.5%

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

   3. pow1/2 12.5%

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

   4. pow-prod-down 12.6%

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

7. Applied egg-rr 12.6%

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

8. Final simplification 12.6%

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

Alternative 16: 26.9% 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 19.0%

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

3. Taylor expanded in B around inf 12.3%

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

   1. mul-1-neg 12.3%

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

   2. *-commutative 12.3%

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

5. Simplified 12.3%

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

   1. *-commutative 12.3%

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

   2. pow1/2 12.5%

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

   3. pow1/2 12.5%

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

   4. pow-prod-down 12.6%

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

7. Applied egg-rr 12.6%

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

   1. unpow1/2 12.3%

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

9. Simplified 12.3%

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

10. Final simplification 12.3%

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

Alternative 17: 26.9% 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 19.0%

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

3. Taylor expanded in B around inf 12.3%

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

   1. mul-1-neg 12.3%

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

   2. *-commutative 12.3%

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

5. Simplified 12.3%

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

   1. pow1/2 12.5%

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

   2. div-inv 12.5%

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

   3. unpow-prod-down 16.5%

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

   4. pow1/2 16.5%

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

7. Applied egg-rr 16.5%

   \[\leadsto -\sqrt{2} \cdot \color{blue}{\left(\sqrt{F} \cdot {\left(\frac{1}{B}\right)}^{0.5}\right)} \]
8. Step-by-step derivation

   1. unpow1/2 16.5%

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

9. Simplified 16.5%

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

    1. neg-sub0 16.5%

       \[\leadsto \color{blue}{0 - \sqrt{2} \cdot \left(\sqrt{F} \cdot \sqrt{\frac{1}{B}}\right)} \]

    2. sqrt-unprod 12.3%

       \[\leadsto 0 - \sqrt{2} \cdot \color{blue}{\sqrt{F \cdot \frac{1}{B}}} \]

    3. div-inv 12.3%

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

    4. sqrt-prod 12.3%

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

    5. clear-num 12.4%

       \[\leadsto 0 - \sqrt{2 \cdot \color{blue}{\frac{1}{\frac{B}{F}}}} \]

    6. un-div-inv 12.4%

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

11. Applied egg-rr 12.4%

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

    1. neg-sub0 12.4%

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

    2. associate-/r/ 12.3%

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

13. Simplified 12.3%

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

14. Final simplification 12.3%

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

Alternative 18: 2.4% accurate, 6.0× speedup

Math

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

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 (/ B_m F))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return sqrt((2.0 / (B_m / F)));
}

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 / (b_m / f)))
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 / (B_m / F)));
}

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 / (B_m / F)))

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return sqrt(Float64(2.0 / Float64(B_m / F)))
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 / (B_m / F)));
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[(B$95$m / F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 19.0%

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

3. Taylor expanded in B around inf 12.3%

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

   1. mul-1-neg 12.3%

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

   2. *-commutative 12.3%

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

5. Simplified 12.3%

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

   1. pow1/2 12.5%

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

   2. div-inv 12.5%

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

   3. unpow-prod-down 16.5%

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

   4. pow1/2 16.5%

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

7. Applied egg-rr 16.5%

   \[\leadsto -\sqrt{2} \cdot \color{blue}{\left(\sqrt{F} \cdot {\left(\frac{1}{B}\right)}^{0.5}\right)} \]
8. Step-by-step derivation

   1. unpow1/2 16.5%

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

9. Simplified 16.5%

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

    1. pow1/2 16.5%

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

    2. exp-to-pow 16.5%

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

    3. distribute-lft-neg-in 16.5%

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

    4. sqrt-unprod 12.3%

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

    5. div-inv 12.3%

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

    6. distribute-lft-neg-in 12.3%

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

    7. add-sqr-sqrt 0.6%

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

    8. sqrt-unprod 1.9%

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

    9. sqr-neg 1.9%

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

    10. sqrt-prod 1.9%

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

    11. add-sqr-sqrt 1.9%

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

    12. exp-to-pow 1.9%

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

    13. pow1/2 1.9%

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

    14. sqrt-prod 1.9%

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

    15. clear-num 2.0%

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

    16. un-div-inv 2.0%

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

11. Applied egg-rr 2.0%

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

Alternative 19: 2.4% accurate, 6.0× speedup

Math

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

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 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 19.0%

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

3. Taylor expanded in B around inf 12.3%

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

   1. mul-1-neg 12.3%

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

   2. *-commutative 12.3%

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

5. Simplified 12.3%

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

   1. pow1/2 12.5%

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

   2. div-inv 12.5%

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

   3. unpow-prod-down 16.5%

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

   4. pow1/2 16.5%

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

7. Applied egg-rr 16.5%

   \[\leadsto -\sqrt{2} \cdot \color{blue}{\left(\sqrt{F} \cdot {\left(\frac{1}{B}\right)}^{0.5}\right)} \]
8. Step-by-step derivation

   1. unpow1/2 16.5%

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

9. Simplified 16.5%

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

    1. pow1/2 16.5%

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

    2. exp-to-pow 16.5%

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

    3. distribute-lft-neg-in 16.5%

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

    4. sqrt-unprod 12.3%

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

    5. div-inv 12.3%

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

    6. distribute-lft-neg-in 12.3%

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

    7. neg-sub0 12.3%

       \[\leadsto \color{blue}{0 - e^{\log 2 \cdot 0.5} \cdot \sqrt{\frac{F}{B}}} \]

    8. sub-neg 12.3%

       \[\leadsto \color{blue}{0 + \left(-e^{\log 2 \cdot 0.5} \cdot \sqrt{\frac{F}{B}}\right)} \]

    9. add-sqr-sqrt 0.6%

       \[\leadsto 0 + \color{blue}{\sqrt{-e^{\log 2 \cdot 0.5} \cdot \sqrt{\frac{F}{B}}} \cdot \sqrt{-e^{\log 2 \cdot 0.5} \cdot \sqrt{\frac{F}{B}}}} \]

    10. sqrt-unprod 1.9%

        \[\leadsto 0 + \color{blue}{\sqrt{\left(-e^{\log 2 \cdot 0.5} \cdot \sqrt{\frac{F}{B}}\right) \cdot \left(-e^{\log 2 \cdot 0.5} \cdot \sqrt{\frac{F}{B}}\right)}} \]

    11. sqr-neg 1.9%

        \[\leadsto 0 + \sqrt{\color{blue}{\left(e^{\log 2 \cdot 0.5} \cdot \sqrt{\frac{F}{B}}\right) \cdot \left(e^{\log 2 \cdot 0.5} \cdot \sqrt{\frac{F}{B}}\right)}} \]

    12. sqrt-prod 1.9%

        \[\leadsto 0 + \color{blue}{\sqrt{e^{\log 2 \cdot 0.5} \cdot \sqrt{\frac{F}{B}}} \cdot \sqrt{e^{\log 2 \cdot 0.5} \cdot \sqrt{\frac{F}{B}}}} \]

    13. add-sqr-sqrt 1.9%

        \[\leadsto 0 + \color{blue}{e^{\log 2 \cdot 0.5} \cdot \sqrt{\frac{F}{B}}} \]

11. Applied egg-rr 2.0%

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

    1. +-lft-identity 2.0%

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

    2. associate-/r/ 1.9%

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

13. Simplified 1.9%

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

14. Final simplification 1.9%

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

Reproduce

herbie shell --seed 2024146 
(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))))