ABCF->ab-angle a

Percentage Accurate: 19.4% → 47.3%

Time: 31.6s

Alternatives: 18

Speedup: 3.1×

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: 19.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 47.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(A - C, B_m\right)\\ t_1 := \mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B_m \leq 2.9 \cdot 10^{-201}:\\ \;\;\;\;\frac{\sqrt{\left(C + \left(A + t_0\right)\right) \cdot t_1} \cdot \left(-\sqrt{2 \cdot F}\right)}{\mathsf{fma}\left(B_m, B_m, -4 \cdot \left(C \cdot A\right)\right)}\\ \mathbf{elif}\;B_m \leq 5 \cdot 10^{-161}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left({B_m}^{2} - \left(C \cdot A\right) \cdot 4\right)\right)} \cdot \left(-\sqrt{2 \cdot C}\right)}{{B_m}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;B_m \leq 3.05 \cdot 10^{+68}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot t_1\right) \cdot \left(F \cdot \left(A + \left(C + t_0\right)\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(B_m, C\right)} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{2}}{B_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (hypot (- A C) B_m)) (t_1 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= B_m 2.9e-201)
     (/
      (* (sqrt (* (+ C (+ A t_0)) t_1)) (- (sqrt (* 2.0 F))))
      (fma B_m B_m (* -4.0 (* C A))))
     (if (<= B_m 5e-161)
       (/
        (*
         (sqrt (* 2.0 (* F (- (pow B_m 2.0) (* (* C A) 4.0)))))
         (- (sqrt (* 2.0 C))))
        (- (pow B_m 2.0) (* C (* A 4.0))))
       (if (<= B_m 3.05e+68)
         (/ (- (sqrt (* (* 2.0 t_1) (* F (+ A (+ C t_0)))))) t_1)
         (*
          (* (sqrt (+ C (hypot B_m C))) (sqrt F))
          (/ (- (sqrt 2.0)) B_m)))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = hypot((A - C), B_m);
	double t_1 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (B_m <= 2.9e-201) {
		tmp = (sqrt(((C + (A + t_0)) * t_1)) * -sqrt((2.0 * F))) / fma(B_m, B_m, (-4.0 * (C * A)));
	} else if (B_m <= 5e-161) {
		tmp = (sqrt((2.0 * (F * (pow(B_m, 2.0) - ((C * A) * 4.0))))) * -sqrt((2.0 * C))) / (pow(B_m, 2.0) - (C * (A * 4.0)));
	} else if (B_m <= 3.05e+68) {
		tmp = -sqrt(((2.0 * t_1) * (F * (A + (C + t_0))))) / t_1;
	} else {
		tmp = (sqrt((C + hypot(B_m, C))) * sqrt(F)) * (-sqrt(2.0) / B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = hypot(Float64(A - C), B_m)
	t_1 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if (B_m <= 2.9e-201)
		tmp = Float64(Float64(sqrt(Float64(Float64(C + Float64(A + t_0)) * t_1)) * Float64(-sqrt(Float64(2.0 * F)))) / fma(B_m, B_m, Float64(-4.0 * Float64(C * A))));
	elseif (B_m <= 5e-161)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(F * Float64((B_m ^ 2.0) - Float64(Float64(C * A) * 4.0))))) * Float64(-sqrt(Float64(2.0 * C)))) / Float64((B_m ^ 2.0) - Float64(C * Float64(A * 4.0))));
	elseif (B_m <= 3.05e+68)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * t_1) * Float64(F * Float64(A + Float64(C + t_0)))))) / t_1);
	else
		tmp = Float64(Float64(sqrt(Float64(C + hypot(B_m, C))) * sqrt(F)) * Float64(Float64(-sqrt(2.0)) / B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]}, Block[{t$95$1 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 2.9e-201], N[(N[(N[Sqrt[N[(N[(C + N[(A + t$95$0), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / N[(B$95$m * B$95$m + N[(-4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 5e-161], N[(N[(N[Sqrt[N[(2.0 * N[(F * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(C * A), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 3.05e+68], N[((-N[Sqrt[N[(N[(2.0 * t$95$1), $MachinePrecision] * N[(F * N[(A + N[(C + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], N[(N[(N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 2: 43.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := {B_m}^{2} - C \cdot \left(A \cdot 4\right)\\ t_2 := 2 \cdot \left(F \cdot \left({B_m}^{2} - \left(C \cdot A\right) \cdot 4\right)\right)\\ \mathbf{if}\;{B_m}^{2} \leq 1.9 \cdot 10^{-293}:\\ \;\;\;\;\frac{{\left(2 \cdot t_2\right)}^{0.5} \cdot \left(-\sqrt{C}\right)}{t_1}\\ \mathbf{elif}\;{B_m}^{2} \leq 10^{-131}:\\ \;\;\;\;\frac{-\sqrt{\left(t_0 \cdot \left(2 \cdot F\right)\right) \cdot \left(A + A\right)}}{t_0}\\ \mathbf{elif}\;{B_m}^{2} \leq 2 \cdot 10^{+19}:\\ \;\;\;\;\frac{\sqrt{t_2} \cdot \left(-\sqrt{2 \cdot C}\right)}{t_1}\\ \mathbf{elif}\;{B_m}^{2} \leq 2 \cdot 10^{+83}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot t_1\right)\right) \cdot \left(A + \mathsf{hypot}\left(B_m, A\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(B_m, C\right)} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{2}}{B_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0))))
        (t_1 (- (pow B_m 2.0) (* C (* A 4.0))))
        (t_2 (* 2.0 (* F (- (pow B_m 2.0) (* (* C A) 4.0))))))
   (if (<= (pow B_m 2.0) 1.9e-293)
     (/ (* (pow (* 2.0 t_2) 0.5) (- (sqrt C))) t_1)
     (if (<= (pow B_m 2.0) 1e-131)
       (/ (- (sqrt (* (* t_0 (* 2.0 F)) (+ A A)))) t_0)
       (if (<= (pow B_m 2.0) 2e+19)
         (/ (* (sqrt t_2) (- (sqrt (* 2.0 C)))) t_1)
         (if (<= (pow B_m 2.0) 2e+83)
           (/ (- (sqrt (* (* 2.0 (* F t_1)) (+ A (hypot B_m A))))) t_1)
           (*
            (* (sqrt (+ C (hypot B_m C))) (sqrt F))
            (/ (- (sqrt 2.0)) B_m))))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_1 = pow(B_m, 2.0) - (C * (A * 4.0));
	double t_2 = 2.0 * (F * (pow(B_m, 2.0) - ((C * A) * 4.0)));
	double tmp;
	if (pow(B_m, 2.0) <= 1.9e-293) {
		tmp = (pow((2.0 * t_2), 0.5) * -sqrt(C)) / t_1;
	} else if (pow(B_m, 2.0) <= 1e-131) {
		tmp = -sqrt(((t_0 * (2.0 * F)) * (A + A))) / t_0;
	} else if (pow(B_m, 2.0) <= 2e+19) {
		tmp = (sqrt(t_2) * -sqrt((2.0 * C))) / t_1;
	} else if (pow(B_m, 2.0) <= 2e+83) {
		tmp = -sqrt(((2.0 * (F * t_1)) * (A + hypot(B_m, A)))) / t_1;
	} else {
		tmp = (sqrt((C + hypot(B_m, C))) * sqrt(F)) * (-sqrt(2.0) / B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_1 = Float64((B_m ^ 2.0) - Float64(C * Float64(A * 4.0)))
	t_2 = Float64(2.0 * Float64(F * Float64((B_m ^ 2.0) - Float64(Float64(C * A) * 4.0))))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1.9e-293)
		tmp = Float64(Float64((Float64(2.0 * t_2) ^ 0.5) * Float64(-sqrt(C))) / t_1);
	elseif ((B_m ^ 2.0) <= 1e-131)
		tmp = Float64(Float64(-sqrt(Float64(Float64(t_0 * Float64(2.0 * F)) * Float64(A + A)))) / t_0);
	elseif ((B_m ^ 2.0) <= 2e+19)
		tmp = Float64(Float64(sqrt(t_2) * Float64(-sqrt(Float64(2.0 * C)))) / t_1);
	elseif ((B_m ^ 2.0) <= 2e+83)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(F * t_1)) * Float64(A + hypot(B_m, A))))) / t_1);
	else
		tmp = Float64(Float64(sqrt(Float64(C + hypot(B_m, C))) * sqrt(F)) * Float64(Float64(-sqrt(2.0)) / B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(F * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(C * A), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1.9e-293], N[(N[(N[Power[N[(2.0 * t$95$2), $MachinePrecision], 0.5], $MachinePrecision] * (-N[Sqrt[C], $MachinePrecision])), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-131], N[((-N[Sqrt[N[(N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision] * N[(A + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+19], N[(N[(N[Sqrt[t$95$2], $MachinePrecision] * (-N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+83], N[((-N[Sqrt[N[(N[(2.0 * N[(F * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], N[(N[(N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 3: 53.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;{B_m}^{2} \leq 2 \cdot 10^{+137}:\\ \;\;\;\;\frac{\sqrt{C + \left(A + \mathsf{hypot}\left(A - C, B_m\right)\right)} \cdot \left(-\sqrt{\mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)}\right)}{\mathsf{fma}\left(B_m, B_m, -4 \cdot \left(C \cdot A\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(B_m, C\right)} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{2}}{B_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 2e+137)
   (/
    (*
     (sqrt (+ C (+ A (hypot (- A C) B_m))))
     (- (sqrt (* (fma B_m B_m (* A (* C -4.0))) (* 2.0 F)))))
    (fma B_m B_m (* -4.0 (* C A))))
   (* (* (sqrt (+ C (hypot B_m C))) (sqrt F)) (/ (- (sqrt 2.0)) B_m))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 2e+137) {
		tmp = (sqrt((C + (A + hypot((A - C), B_m)))) * -sqrt((fma(B_m, B_m, (A * (C * -4.0))) * (2.0 * F)))) / fma(B_m, B_m, (-4.0 * (C * A)));
	} else {
		tmp = (sqrt((C + hypot(B_m, C))) * sqrt(F)) * (-sqrt(2.0) / B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e+137)
		tmp = Float64(Float64(sqrt(Float64(C + Float64(A + hypot(Float64(A - C), B_m)))) * Float64(-sqrt(Float64(fma(B_m, B_m, Float64(A * Float64(C * -4.0))) * Float64(2.0 * F))))) / fma(B_m, B_m, Float64(-4.0 * Float64(C * A))));
	else
		tmp = Float64(Float64(sqrt(Float64(C + hypot(B_m, C))) * sqrt(F)) * Float64(Float64(-sqrt(2.0)) / B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+137], N[(N[(N[Sqrt[N[(C + N[(A + N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / N[(B$95$m * B$95$m + N[(-4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 4: 43.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := {B_m}^{2} - C \cdot \left(A \cdot 4\right)\\ t_2 := \frac{\sqrt{2 \cdot \left(F \cdot \left({B_m}^{2} - \left(C \cdot A\right) \cdot 4\right)\right)} \cdot \left(-\sqrt{2 \cdot C}\right)}{t_1}\\ \mathbf{if}\;B_m \leq 4.4 \cdot 10^{-147}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B_m \leq 4.4 \cdot 10^{-66}:\\ \;\;\;\;\frac{-\sqrt{\left(t_0 \cdot \left(2 \cdot F\right)\right) \cdot \left(A + A\right)}}{t_0}\\ \mathbf{elif}\;B_m \leq 3400000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B_m \leq 7.5 \cdot 10^{+41}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot t_1\right)\right) \cdot \left(A + \mathsf{hypot}\left(B_m, A\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(B_m, C\right)} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{2}}{B_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0))))
        (t_1 (- (pow B_m 2.0) (* C (* A 4.0))))
        (t_2
         (/
          (*
           (sqrt (* 2.0 (* F (- (pow B_m 2.0) (* (* C A) 4.0)))))
           (- (sqrt (* 2.0 C))))
          t_1)))
   (if (<= B_m 4.4e-147)
     t_2
     (if (<= B_m 4.4e-66)
       (/ (- (sqrt (* (* t_0 (* 2.0 F)) (+ A A)))) t_0)
       (if (<= B_m 3400000000.0)
         t_2
         (if (<= B_m 7.5e+41)
           (/ (- (sqrt (* (* 2.0 (* F t_1)) (+ A (hypot B_m A))))) t_1)
           (*
            (* (sqrt (+ C (hypot B_m C))) (sqrt F))
            (/ (- (sqrt 2.0)) B_m))))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_1 = pow(B_m, 2.0) - (C * (A * 4.0));
	double t_2 = (sqrt((2.0 * (F * (pow(B_m, 2.0) - ((C * A) * 4.0))))) * -sqrt((2.0 * C))) / t_1;
	double tmp;
	if (B_m <= 4.4e-147) {
		tmp = t_2;
	} else if (B_m <= 4.4e-66) {
		tmp = -sqrt(((t_0 * (2.0 * F)) * (A + A))) / t_0;
	} else if (B_m <= 3400000000.0) {
		tmp = t_2;
	} else if (B_m <= 7.5e+41) {
		tmp = -sqrt(((2.0 * (F * t_1)) * (A + hypot(B_m, A)))) / t_1;
	} else {
		tmp = (sqrt((C + hypot(B_m, C))) * sqrt(F)) * (-sqrt(2.0) / B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_1 = Float64((B_m ^ 2.0) - Float64(C * Float64(A * 4.0)))
	t_2 = Float64(Float64(sqrt(Float64(2.0 * Float64(F * Float64((B_m ^ 2.0) - Float64(Float64(C * A) * 4.0))))) * Float64(-sqrt(Float64(2.0 * C)))) / t_1)
	tmp = 0.0
	if (B_m <= 4.4e-147)
		tmp = t_2;
	elseif (B_m <= 4.4e-66)
		tmp = Float64(Float64(-sqrt(Float64(Float64(t_0 * Float64(2.0 * F)) * Float64(A + A)))) / t_0);
	elseif (B_m <= 3400000000.0)
		tmp = t_2;
	elseif (B_m <= 7.5e+41)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(F * t_1)) * Float64(A + hypot(B_m, A))))) / t_1);
	else
		tmp = Float64(Float64(sqrt(Float64(C + hypot(B_m, C))) * sqrt(F)) * Float64(Float64(-sqrt(2.0)) / B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sqrt[N[(2.0 * N[(F * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(C * A), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[B$95$m, 4.4e-147], t$95$2, If[LessEqual[B$95$m, 4.4e-66], N[((-N[Sqrt[N[(N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision] * N[(A + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[B$95$m, 3400000000.0], t$95$2, If[LessEqual[B$95$m, 7.5e+41], N[((-N[Sqrt[N[(N[(2.0 * N[(F * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], N[(N[(N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 5: 43.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := {B_m}^{2} - C \cdot \left(A \cdot 4\right)\\ t_2 := \mathsf{fma}\left(B_m, B_m, -4 \cdot \left(C \cdot A\right)\right)\\ \mathbf{if}\;B_m \leq 4.4 \cdot 10^{-147}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)\right) \cdot \left(2 \cdot C\right)}}{t_1}\\ \mathbf{elif}\;B_m \leq 5 \cdot 10^{-66}:\\ \;\;\;\;\frac{-\sqrt{\left(t_0 \cdot \left(2 \cdot F\right)\right) \cdot \left(A + A\right)}}{t_0}\\ \mathbf{elif}\;B_m \leq 35000000000:\\ \;\;\;\;\frac{-\sqrt{4 \cdot \left(C \cdot \left(F \cdot \left({B_m}^{2} - \left(C \cdot A\right) \cdot 4\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;B_m \leq 7.5 \cdot 10^{+41}:\\ \;\;\;\;\frac{-\sqrt{\left(t_2 \cdot \left(A + \mathsf{hypot}\left(B_m, A\right)\right)\right) \cdot \left(2 \cdot F\right)}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(B_m, C\right)} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{2}}{B_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0))))
        (t_1 (- (pow B_m 2.0) (* C (* A 4.0))))
        (t_2 (fma B_m B_m (* -4.0 (* C A)))))
   (if (<= B_m 4.4e-147)
     (/ (- (sqrt (* (* 2.0 (* -4.0 (* A (* C F)))) (* 2.0 C)))) t_1)
     (if (<= B_m 5e-66)
       (/ (- (sqrt (* (* t_0 (* 2.0 F)) (+ A A)))) t_0)
       (if (<= B_m 35000000000.0)
         (/
          (- (sqrt (* 4.0 (* C (* F (- (pow B_m 2.0) (* (* C A) 4.0)))))))
          t_1)
         (if (<= B_m 7.5e+41)
           (/ (- (sqrt (* (* t_2 (+ A (hypot B_m A))) (* 2.0 F)))) t_2)
           (*
            (* (sqrt (+ C (hypot B_m C))) (sqrt F))
            (/ (- (sqrt 2.0)) B_m))))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_1 = pow(B_m, 2.0) - (C * (A * 4.0));
	double t_2 = fma(B_m, B_m, (-4.0 * (C * A)));
	double tmp;
	if (B_m <= 4.4e-147) {
		tmp = -sqrt(((2.0 * (-4.0 * (A * (C * F)))) * (2.0 * C))) / t_1;
	} else if (B_m <= 5e-66) {
		tmp = -sqrt(((t_0 * (2.0 * F)) * (A + A))) / t_0;
	} else if (B_m <= 35000000000.0) {
		tmp = -sqrt((4.0 * (C * (F * (pow(B_m, 2.0) - ((C * A) * 4.0)))))) / t_1;
	} else if (B_m <= 7.5e+41) {
		tmp = -sqrt(((t_2 * (A + hypot(B_m, A))) * (2.0 * F))) / t_2;
	} else {
		tmp = (sqrt((C + hypot(B_m, C))) * sqrt(F)) * (-sqrt(2.0) / B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_1 = Float64((B_m ^ 2.0) - Float64(C * Float64(A * 4.0)))
	t_2 = fma(B_m, B_m, Float64(-4.0 * Float64(C * A)))
	tmp = 0.0
	if (B_m <= 4.4e-147)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(-4.0 * Float64(A * Float64(C * F)))) * Float64(2.0 * C)))) / t_1);
	elseif (B_m <= 5e-66)
		tmp = Float64(Float64(-sqrt(Float64(Float64(t_0 * Float64(2.0 * F)) * Float64(A + A)))) / t_0);
	elseif (B_m <= 35000000000.0)
		tmp = Float64(Float64(-sqrt(Float64(4.0 * Float64(C * Float64(F * Float64((B_m ^ 2.0) - Float64(Float64(C * A) * 4.0))))))) / t_1);
	elseif (B_m <= 7.5e+41)
		tmp = Float64(Float64(-sqrt(Float64(Float64(t_2 * Float64(A + hypot(B_m, A))) * Float64(2.0 * F)))) / t_2);
	else
		tmp = Float64(Float64(sqrt(Float64(C + hypot(B_m, C))) * sqrt(F)) * Float64(Float64(-sqrt(2.0)) / B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(B$95$m * B$95$m + N[(-4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 4.4e-147], N[((-N[Sqrt[N[(N[(2.0 * N[(-4.0 * N[(A * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], If[LessEqual[B$95$m, 5e-66], N[((-N[Sqrt[N[(N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision] * N[(A + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[B$95$m, 35000000000.0], N[((-N[Sqrt[N[(4.0 * N[(C * N[(F * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(C * A), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], If[LessEqual[B$95$m, 7.5e+41], N[((-N[Sqrt[N[(N[(t$95$2 * N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision], N[(N[(N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 6: 43.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \mathsf{fma}\left(B_m, B_m, -4 \cdot \left(C \cdot A\right)\right)\\ t_2 := \frac{\sqrt{2 \cdot \left(F \cdot \left({B_m}^{2} - \left(C \cdot A\right) \cdot 4\right)\right)} \cdot \left(-\sqrt{2 \cdot C}\right)}{{B_m}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{if}\;B_m \leq 4.4 \cdot 10^{-147}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B_m \leq 5.6 \cdot 10^{-66}:\\ \;\;\;\;\frac{-\sqrt{\left(t_0 \cdot \left(2 \cdot F\right)\right) \cdot \left(A + A\right)}}{t_0}\\ \mathbf{elif}\;B_m \leq 140000000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B_m \leq 7.5 \cdot 10^{+41}:\\ \;\;\;\;\frac{-\sqrt{\left(t_1 \cdot \left(A + \mathsf{hypot}\left(B_m, A\right)\right)\right) \cdot \left(2 \cdot F\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(B_m, C\right)} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{2}}{B_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0))))
        (t_1 (fma B_m B_m (* -4.0 (* C A))))
        (t_2
         (/
          (*
           (sqrt (* 2.0 (* F (- (pow B_m 2.0) (* (* C A) 4.0)))))
           (- (sqrt (* 2.0 C))))
          (- (pow B_m 2.0) (* C (* A 4.0))))))
   (if (<= B_m 4.4e-147)
     t_2
     (if (<= B_m 5.6e-66)
       (/ (- (sqrt (* (* t_0 (* 2.0 F)) (+ A A)))) t_0)
       (if (<= B_m 140000000000.0)
         t_2
         (if (<= B_m 7.5e+41)
           (/ (- (sqrt (* (* t_1 (+ A (hypot B_m A))) (* 2.0 F)))) t_1)
           (*
            (* (sqrt (+ C (hypot B_m C))) (sqrt F))
            (/ (- (sqrt 2.0)) B_m))))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_1 = fma(B_m, B_m, (-4.0 * (C * A)));
	double t_2 = (sqrt((2.0 * (F * (pow(B_m, 2.0) - ((C * A) * 4.0))))) * -sqrt((2.0 * C))) / (pow(B_m, 2.0) - (C * (A * 4.0)));
	double tmp;
	if (B_m <= 4.4e-147) {
		tmp = t_2;
	} else if (B_m <= 5.6e-66) {
		tmp = -sqrt(((t_0 * (2.0 * F)) * (A + A))) / t_0;
	} else if (B_m <= 140000000000.0) {
		tmp = t_2;
	} else if (B_m <= 7.5e+41) {
		tmp = -sqrt(((t_1 * (A + hypot(B_m, A))) * (2.0 * F))) / t_1;
	} else {
		tmp = (sqrt((C + hypot(B_m, C))) * sqrt(F)) * (-sqrt(2.0) / B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_1 = fma(B_m, B_m, Float64(-4.0 * Float64(C * A)))
	t_2 = Float64(Float64(sqrt(Float64(2.0 * Float64(F * Float64((B_m ^ 2.0) - Float64(Float64(C * A) * 4.0))))) * Float64(-sqrt(Float64(2.0 * C)))) / Float64((B_m ^ 2.0) - Float64(C * Float64(A * 4.0))))
	tmp = 0.0
	if (B_m <= 4.4e-147)
		tmp = t_2;
	elseif (B_m <= 5.6e-66)
		tmp = Float64(Float64(-sqrt(Float64(Float64(t_0 * Float64(2.0 * F)) * Float64(A + A)))) / t_0);
	elseif (B_m <= 140000000000.0)
		tmp = t_2;
	elseif (B_m <= 7.5e+41)
		tmp = Float64(Float64(-sqrt(Float64(Float64(t_1 * Float64(A + hypot(B_m, A))) * Float64(2.0 * F)))) / t_1);
	else
		tmp = Float64(Float64(sqrt(Float64(C + hypot(B_m, C))) * sqrt(F)) * Float64(Float64(-sqrt(2.0)) / B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(B$95$m * B$95$m + N[(-4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sqrt[N[(2.0 * N[(F * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(C * A), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 4.4e-147], t$95$2, If[LessEqual[B$95$m, 5.6e-66], N[((-N[Sqrt[N[(N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision] * N[(A + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[B$95$m, 140000000000.0], t$95$2, If[LessEqual[B$95$m, 7.5e+41], N[((-N[Sqrt[N[(N[(t$95$1 * N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], N[(N[(N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 7: 48.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B_m \leq 1.82 \cdot 10^{-162}:\\ \;\;\;\;\frac{{\left(2 \cdot \left(2 \cdot \left(F \cdot \left({B_m}^{2} - \left(C \cdot A\right) \cdot 4\right)\right)\right)\right)}^{0.5} \cdot \left(-\sqrt{C}\right)}{{B_m}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;B_m \leq 3 \cdot 10^{+68}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot t_0\right) \cdot \left(F \cdot \left(A + \left(C + \mathsf{hypot}\left(A - C, B_m\right)\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(B_m, C\right)} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{2}}{B_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= B_m 1.82e-162)
     (/
      (*
       (pow (* 2.0 (* 2.0 (* F (- (pow B_m 2.0) (* (* C A) 4.0))))) 0.5)
       (- (sqrt C)))
      (- (pow B_m 2.0) (* C (* A 4.0))))
     (if (<= B_m 3e+68)
       (/ (- (sqrt (* (* 2.0 t_0) (* F (+ A (+ C (hypot (- A C) B_m))))))) t_0)
       (* (* (sqrt (+ C (hypot B_m C))) (sqrt F)) (/ (- (sqrt 2.0)) B_m))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (B_m <= 1.82e-162) {
		tmp = (pow((2.0 * (2.0 * (F * (pow(B_m, 2.0) - ((C * A) * 4.0))))), 0.5) * -sqrt(C)) / (pow(B_m, 2.0) - (C * (A * 4.0)));
	} else if (B_m <= 3e+68) {
		tmp = -sqrt(((2.0 * t_0) * (F * (A + (C + hypot((A - C), B_m)))))) / t_0;
	} else {
		tmp = (sqrt((C + hypot(B_m, C))) * sqrt(F)) * (-sqrt(2.0) / B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if (B_m <= 1.82e-162)
		tmp = Float64(Float64((Float64(2.0 * Float64(2.0 * Float64(F * Float64((B_m ^ 2.0) - Float64(Float64(C * A) * 4.0))))) ^ 0.5) * Float64(-sqrt(C))) / Float64((B_m ^ 2.0) - Float64(C * Float64(A * 4.0))));
	elseif (B_m <= 3e+68)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * t_0) * Float64(F * Float64(A + Float64(C + hypot(Float64(A - C), B_m))))))) / t_0);
	else
		tmp = Float64(Float64(sqrt(Float64(C + hypot(B_m, C))) * sqrt(F)) * Float64(Float64(-sqrt(2.0)) / B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 1.82e-162], N[(N[(N[Power[N[(2.0 * N[(2.0 * N[(F * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(C * A), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] * (-N[Sqrt[C], $MachinePrecision])), $MachinePrecision] / N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 3e+68], N[((-N[Sqrt[N[(N[(2.0 * t$95$0), $MachinePrecision] * N[(F * N[(A + N[(C + N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision]), $MachinePrecision]]]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 8: 43.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := -\sqrt{2}\\ t_2 := {B_m}^{2} - C \cdot \left(A \cdot 4\right)\\ t_3 := \frac{t_1}{B_m}\\ \mathbf{if}\;B_m \leq 4.4 \cdot 10^{-147}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)\right) \cdot \left(2 \cdot C\right)}}{t_2}\\ \mathbf{elif}\;B_m \leq 3.6 \cdot 10^{-66}:\\ \;\;\;\;\frac{-\sqrt{\left(t_0 \cdot \left(2 \cdot F\right)\right) \cdot \left(A + A\right)}}{t_0}\\ \mathbf{elif}\;B_m \leq 4200000000:\\ \;\;\;\;\frac{-\sqrt{4 \cdot \left(C \cdot \left(F \cdot \left({B_m}^{2} - \left(C \cdot A\right) \cdot 4\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;B_m \leq 2.9 \cdot 10^{+39}:\\ \;\;\;\;\frac{B_m \cdot \left(\sqrt{F \cdot \left(A + \mathsf{hypot}\left(B_m, A\right)\right)} \cdot t_1\right)}{\mathsf{fma}\left(B_m, B_m, -4 \cdot \left(C \cdot A\right)\right)}\\ \mathbf{elif}\;B_m \leq 7.4 \cdot 10^{+45}:\\ \;\;\;\;t_3 \cdot \sqrt{F \cdot \left(\frac{{B_m}^{2}}{C} \cdot -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(B_m, C\right)} \cdot \sqrt{F}\right) \cdot t_3\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0))))
        (t_1 (- (sqrt 2.0)))
        (t_2 (- (pow B_m 2.0) (* C (* A 4.0))))
        (t_3 (/ t_1 B_m)))
   (if (<= B_m 4.4e-147)
     (/ (- (sqrt (* (* 2.0 (* -4.0 (* A (* C F)))) (* 2.0 C)))) t_2)
     (if (<= B_m 3.6e-66)
       (/ (- (sqrt (* (* t_0 (* 2.0 F)) (+ A A)))) t_0)
       (if (<= B_m 4200000000.0)
         (/
          (- (sqrt (* 4.0 (* C (* F (- (pow B_m 2.0) (* (* C A) 4.0)))))))
          t_2)
         (if (<= B_m 2.9e+39)
           (/
            (* B_m (* (sqrt (* F (+ A (hypot B_m A)))) t_1))
            (fma B_m B_m (* -4.0 (* C A))))
           (if (<= B_m 7.4e+45)
             (* t_3 (sqrt (* F (* (/ (pow B_m 2.0) C) -0.5))))
             (* (* (sqrt (+ C (hypot B_m C))) (sqrt F)) t_3))))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_1 = -sqrt(2.0);
	double t_2 = pow(B_m, 2.0) - (C * (A * 4.0));
	double t_3 = t_1 / B_m;
	double tmp;
	if (B_m <= 4.4e-147) {
		tmp = -sqrt(((2.0 * (-4.0 * (A * (C * F)))) * (2.0 * C))) / t_2;
	} else if (B_m <= 3.6e-66) {
		tmp = -sqrt(((t_0 * (2.0 * F)) * (A + A))) / t_0;
	} else if (B_m <= 4200000000.0) {
		tmp = -sqrt((4.0 * (C * (F * (pow(B_m, 2.0) - ((C * A) * 4.0)))))) / t_2;
	} else if (B_m <= 2.9e+39) {
		tmp = (B_m * (sqrt((F * (A + hypot(B_m, A)))) * t_1)) / fma(B_m, B_m, (-4.0 * (C * A)));
	} else if (B_m <= 7.4e+45) {
		tmp = t_3 * sqrt((F * ((pow(B_m, 2.0) / C) * -0.5)));
	} else {
		tmp = (sqrt((C + hypot(B_m, C))) * sqrt(F)) * t_3;
	}
	return tmp;
}

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_1 = Float64(-sqrt(2.0))
	t_2 = Float64((B_m ^ 2.0) - Float64(C * Float64(A * 4.0)))
	t_3 = Float64(t_1 / B_m)
	tmp = 0.0
	if (B_m <= 4.4e-147)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(-4.0 * Float64(A * Float64(C * F)))) * Float64(2.0 * C)))) / t_2);
	elseif (B_m <= 3.6e-66)
		tmp = Float64(Float64(-sqrt(Float64(Float64(t_0 * Float64(2.0 * F)) * Float64(A + A)))) / t_0);
	elseif (B_m <= 4200000000.0)
		tmp = Float64(Float64(-sqrt(Float64(4.0 * Float64(C * Float64(F * Float64((B_m ^ 2.0) - Float64(Float64(C * A) * 4.0))))))) / t_2);
	elseif (B_m <= 2.9e+39)
		tmp = Float64(Float64(B_m * Float64(sqrt(Float64(F * Float64(A + hypot(B_m, A)))) * t_1)) / fma(B_m, B_m, Float64(-4.0 * Float64(C * A))));
	elseif (B_m <= 7.4e+45)
		tmp = Float64(t_3 * sqrt(Float64(F * Float64(Float64((B_m ^ 2.0) / C) * -0.5))));
	else
		tmp = Float64(Float64(sqrt(Float64(C + hypot(B_m, C))) * sqrt(F)) * t_3);
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[Sqrt[2.0], $MachinePrecision])}, Block[{t$95$2 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 / B$95$m), $MachinePrecision]}, If[LessEqual[B$95$m, 4.4e-147], N[((-N[Sqrt[N[(N[(2.0 * N[(-4.0 * N[(A * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision], If[LessEqual[B$95$m, 3.6e-66], N[((-N[Sqrt[N[(N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision] * N[(A + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[B$95$m, 4200000000.0], N[((-N[Sqrt[N[(4.0 * N[(C * N[(F * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(C * A), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision], If[LessEqual[B$95$m, 2.9e+39], N[(N[(B$95$m * N[(N[Sqrt[N[(F * N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(B$95$m * B$95$m + N[(-4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 7.4e+45], N[(t$95$3 * N[Sqrt[N[(F * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]]]]]]]]]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 9: 43.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := {B_m}^{2} - C \cdot \left(A \cdot 4\right)\\ t_1 := \mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)\\ t_2 := \frac{-\sqrt{2}}{B_m}\\ \mathbf{if}\;B_m \leq 4.4 \cdot 10^{-147}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)\right) \cdot \left(2 \cdot C\right)}}{t_0}\\ \mathbf{elif}\;B_m \leq 4.4 \cdot 10^{-66}:\\ \;\;\;\;\frac{-\sqrt{\left(t_1 \cdot \left(2 \cdot F\right)\right) \cdot \left(A + A\right)}}{t_1}\\ \mathbf{elif}\;B_m \leq 3700000000:\\ \;\;\;\;\frac{-\sqrt{4 \cdot \left(C \cdot \left(F \cdot \left({B_m}^{2} - \left(C \cdot A\right) \cdot 4\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B_m \leq 2.2 \cdot 10^{+39}:\\ \;\;\;\;\frac{\sqrt{2}}{B_m} \cdot \left(-\sqrt{F \cdot \left(A + \mathsf{hypot}\left(B_m, A\right)\right)}\right)\\ \mathbf{elif}\;B_m \leq 7.4 \cdot 10^{+45}:\\ \;\;\;\;t_2 \cdot \sqrt{F \cdot \left(\frac{{B_m}^{2}}{C} \cdot -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(B_m, C\right)} \cdot \sqrt{F}\right) \cdot t_2\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (pow B_m 2.0) (* C (* A 4.0))))
        (t_1 (fma B_m B_m (* A (* C -4.0))))
        (t_2 (/ (- (sqrt 2.0)) B_m)))
   (if (<= B_m 4.4e-147)
     (/ (- (sqrt (* (* 2.0 (* -4.0 (* A (* C F)))) (* 2.0 C)))) t_0)
     (if (<= B_m 4.4e-66)
       (/ (- (sqrt (* (* t_1 (* 2.0 F)) (+ A A)))) t_1)
       (if (<= B_m 3700000000.0)
         (/
          (- (sqrt (* 4.0 (* C (* F (- (pow B_m 2.0) (* (* C A) 4.0)))))))
          t_0)
         (if (<= B_m 2.2e+39)
           (* (/ (sqrt 2.0) B_m) (- (sqrt (* F (+ A (hypot B_m A))))))
           (if (<= B_m 7.4e+45)
             (* t_2 (sqrt (* F (* (/ (pow B_m 2.0) C) -0.5))))
             (* (* (sqrt (+ C (hypot B_m C))) (sqrt F)) t_2))))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = pow(B_m, 2.0) - (C * (A * 4.0));
	double t_1 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_2 = -sqrt(2.0) / B_m;
	double tmp;
	if (B_m <= 4.4e-147) {
		tmp = -sqrt(((2.0 * (-4.0 * (A * (C * F)))) * (2.0 * C))) / t_0;
	} else if (B_m <= 4.4e-66) {
		tmp = -sqrt(((t_1 * (2.0 * F)) * (A + A))) / t_1;
	} else if (B_m <= 3700000000.0) {
		tmp = -sqrt((4.0 * (C * (F * (pow(B_m, 2.0) - ((C * A) * 4.0)))))) / t_0;
	} else if (B_m <= 2.2e+39) {
		tmp = (sqrt(2.0) / B_m) * -sqrt((F * (A + hypot(B_m, A))));
	} else if (B_m <= 7.4e+45) {
		tmp = t_2 * sqrt((F * ((pow(B_m, 2.0) / C) * -0.5)));
	} else {
		tmp = (sqrt((C + hypot(B_m, C))) * sqrt(F)) * t_2;
	}
	return tmp;
}

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64((B_m ^ 2.0) - Float64(C * Float64(A * 4.0)))
	t_1 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_2 = Float64(Float64(-sqrt(2.0)) / B_m)
	tmp = 0.0
	if (B_m <= 4.4e-147)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(-4.0 * Float64(A * Float64(C * F)))) * Float64(2.0 * C)))) / t_0);
	elseif (B_m <= 4.4e-66)
		tmp = Float64(Float64(-sqrt(Float64(Float64(t_1 * Float64(2.0 * F)) * Float64(A + A)))) / t_1);
	elseif (B_m <= 3700000000.0)
		tmp = Float64(Float64(-sqrt(Float64(4.0 * Float64(C * Float64(F * Float64((B_m ^ 2.0) - Float64(Float64(C * A) * 4.0))))))) / t_0);
	elseif (B_m <= 2.2e+39)
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(-sqrt(Float64(F * Float64(A + hypot(B_m, A))))));
	elseif (B_m <= 7.4e+45)
		tmp = Float64(t_2 * sqrt(Float64(F * Float64(Float64((B_m ^ 2.0) / C) * -0.5))));
	else
		tmp = Float64(Float64(sqrt(Float64(C + hypot(B_m, C))) * sqrt(F)) * t_2);
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision]}, If[LessEqual[B$95$m, 4.4e-147], N[((-N[Sqrt[N[(N[(2.0 * N[(-4.0 * N[(A * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[B$95$m, 4.4e-66], N[((-N[Sqrt[N[(N[(t$95$1 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision] * N[(A + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], If[LessEqual[B$95$m, 3700000000.0], N[((-N[Sqrt[N[(4.0 * N[(C * N[(F * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(C * A), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[B$95$m, 2.2e+39], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[N[(F * N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[B$95$m, 7.4e+45], N[(t$95$2 * N[Sqrt[N[(F * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]]]]]]]]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 10: 39.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B_m, B_m, -4 \cdot \left(C \cdot A\right)\right)\\ \mathbf{if}\;F \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot F\right) \cdot \left(t_0 \cdot \left(2 \cdot A\right)\right)}}{t_0}\\ \mathbf{elif}\;F \leq 7.2 \cdot 10^{+22}:\\ \;\;\;\;\frac{\sqrt{2}}{B_m} \cdot \left(-\sqrt{F \cdot \left(A + \mathsf{hypot}\left(B_m, A\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B_m}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* -4.0 (* C A)))))
   (if (<= F -5e-310)
     (/ (- (sqrt (* (* 2.0 F) (* t_0 (* 2.0 A))))) t_0)
     (if (<= F 7.2e+22)
       (* (/ (sqrt 2.0) B_m) (- (sqrt (* F (+ A (hypot B_m A))))))
       (* (sqrt (/ F B_m)) (- (sqrt 2.0)))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (-4.0 * (C * A)));
	double tmp;
	if (F <= -5e-310) {
		tmp = -sqrt(((2.0 * F) * (t_0 * (2.0 * A)))) / t_0;
	} else if (F <= 7.2e+22) {
		tmp = (sqrt(2.0) / B_m) * -sqrt((F * (A + hypot(B_m, A))));
	} else {
		tmp = sqrt((F / B_m)) * -sqrt(2.0);
	}
	return tmp;
}

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(-4.0 * Float64(C * A)))
	tmp = 0.0
	if (F <= -5e-310)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * F) * Float64(t_0 * Float64(2.0 * A))))) / t_0);
	elseif (F <= 7.2e+22)
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(-sqrt(Float64(F * Float64(A + hypot(B_m, A))))));
	else
		tmp = Float64(sqrt(Float64(F / B_m)) * Float64(-sqrt(2.0)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(-4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -5e-310], N[((-N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(t$95$0 * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[F, 7.2e+22], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[N[(F * N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]]]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 11: 38.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;F \leq 6 \cdot 10^{-299}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(\left(C \cdot F\right) \cdot {A}^{2}\right)}}{\mathsf{fma}\left(B_m, B_m, -4 \cdot \left(C \cdot A\right)\right)}\\ \mathbf{elif}\;F \leq 5.8 \cdot 10^{+22}:\\ \;\;\;\;\frac{\sqrt{2}}{B_m} \cdot \left(-\sqrt{F \cdot \left(A + \mathsf{hypot}\left(B_m, A\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B_m}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F 6e-299)
   (/
    (- (sqrt (* -16.0 (* (* C F) (pow A 2.0)))))
    (fma B_m B_m (* -4.0 (* C A))))
   (if (<= F 5.8e+22)
     (* (/ (sqrt 2.0) B_m) (- (sqrt (* F (+ A (hypot B_m A))))))
     (* (sqrt (/ F B_m)) (- (sqrt 2.0))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 6e-299) {
		tmp = -sqrt((-16.0 * ((C * F) * pow(A, 2.0)))) / fma(B_m, B_m, (-4.0 * (C * A)));
	} else if (F <= 5.8e+22) {
		tmp = (sqrt(2.0) / B_m) * -sqrt((F * (A + hypot(B_m, A))));
	} else {
		tmp = sqrt((F / B_m)) * -sqrt(2.0);
	}
	return tmp;
}

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= 6e-299)
		tmp = Float64(Float64(-sqrt(Float64(-16.0 * Float64(Float64(C * F) * (A ^ 2.0))))) / fma(B_m, B_m, Float64(-4.0 * Float64(C * A))));
	elseif (F <= 5.8e+22)
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(-sqrt(Float64(F * Float64(A + hypot(B_m, A))))));
	else
		tmp = Float64(sqrt(Float64(F / B_m)) * Float64(-sqrt(2.0)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, 6e-299], N[((-N[Sqrt[N[(-16.0 * N[(N[(C * F), $MachinePrecision] * N[Power[A, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(B$95$m * B$95$m + N[(-4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5.8e+22], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[N[(F * N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 12: 39.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := -\sqrt{2}\\ \mathbf{if}\;F \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)\right) \cdot \left(2 \cdot C\right)}}{{B_m}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;F \leq 8.5 \cdot 10^{-22}:\\ \;\;\;\;\frac{t_0}{B_m} \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B_m, C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B_m}} \cdot t_0\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (sqrt 2.0))))
   (if (<= F -5e-310)
     (/
      (- (sqrt (* (* 2.0 (* -4.0 (* A (* C F)))) (* 2.0 C))))
      (- (pow B_m 2.0) (* C (* A 4.0))))
     (if (<= F 8.5e-22)
       (* (/ t_0 B_m) (sqrt (* F (+ C (hypot B_m C)))))
       (* (sqrt (/ F B_m)) t_0)))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = -sqrt(2.0);
	double tmp;
	if (F <= -5e-310) {
		tmp = -sqrt(((2.0 * (-4.0 * (A * (C * F)))) * (2.0 * C))) / (pow(B_m, 2.0) - (C * (A * 4.0)));
	} else if (F <= 8.5e-22) {
		tmp = (t_0 / B_m) * sqrt((F * (C + hypot(B_m, C))));
	} else {
		tmp = sqrt((F / B_m)) * t_0;
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double t_0 = -Math.sqrt(2.0);
	double tmp;
	if (F <= -5e-310) {
		tmp = -Math.sqrt(((2.0 * (-4.0 * (A * (C * F)))) * (2.0 * C))) / (Math.pow(B_m, 2.0) - (C * (A * 4.0)));
	} else if (F <= 8.5e-22) {
		tmp = (t_0 / B_m) * Math.sqrt((F * (C + Math.hypot(B_m, C))));
	} else {
		tmp = Math.sqrt((F / B_m)) * t_0;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = -math.sqrt(2.0)
	tmp = 0
	if F <= -5e-310:
		tmp = -math.sqrt(((2.0 * (-4.0 * (A * (C * F)))) * (2.0 * C))) / (math.pow(B_m, 2.0) - (C * (A * 4.0)))
	elif F <= 8.5e-22:
		tmp = (t_0 / B_m) * math.sqrt((F * (C + math.hypot(B_m, C))))
	else:
		tmp = math.sqrt((F / B_m)) * t_0
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(-sqrt(2.0))
	tmp = 0.0
	if (F <= -5e-310)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(-4.0 * Float64(A * Float64(C * F)))) * Float64(2.0 * C)))) / Float64((B_m ^ 2.0) - Float64(C * Float64(A * 4.0))));
	elseif (F <= 8.5e-22)
		tmp = Float64(Float64(t_0 / B_m) * sqrt(Float64(F * Float64(C + hypot(B_m, C)))));
	else
		tmp = Float64(sqrt(Float64(F / B_m)) * t_0);
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = -sqrt(2.0);
	tmp = 0.0;
	if (F <= -5e-310)
		tmp = -sqrt(((2.0 * (-4.0 * (A * (C * F)))) * (2.0 * C))) / ((B_m ^ 2.0) - (C * (A * 4.0)));
	elseif (F <= 8.5e-22)
		tmp = (t_0 / B_m) * sqrt((F * (C + hypot(B_m, C))));
	else
		tmp = sqrt((F / B_m)) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = (-N[Sqrt[2.0], $MachinePrecision])}, If[LessEqual[F, -5e-310], N[((-N[Sqrt[N[(N[(2.0 * N[(-4.0 * N[(A * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 8.5e-22], N[(N[(t$95$0 / B$95$m), $MachinePrecision] * N[Sqrt[N[(F * N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 13: 39.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)\right) \cdot \left(2 \cdot C\right)}}{{B_m}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;F \leq 7.6 \cdot 10^{+22}:\\ \;\;\;\;\frac{\sqrt{2}}{B_m} \cdot \left(-\sqrt{F \cdot \left(A + \mathsf{hypot}\left(B_m, A\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B_m}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F -5e-310)
   (/
    (- (sqrt (* (* 2.0 (* -4.0 (* A (* C F)))) (* 2.0 C))))
    (- (pow B_m 2.0) (* C (* A 4.0))))
   (if (<= F 7.6e+22)
     (* (/ (sqrt 2.0) B_m) (- (sqrt (* F (+ A (hypot B_m A))))))
     (* (sqrt (/ F B_m)) (- (sqrt 2.0))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= -5e-310) {
		tmp = -sqrt(((2.0 * (-4.0 * (A * (C * F)))) * (2.0 * C))) / (pow(B_m, 2.0) - (C * (A * 4.0)));
	} else if (F <= 7.6e+22) {
		tmp = (sqrt(2.0) / B_m) * -sqrt((F * (A + hypot(B_m, A))));
	} else {
		tmp = sqrt((F / B_m)) * -sqrt(2.0);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= -5e-310) {
		tmp = -Math.sqrt(((2.0 * (-4.0 * (A * (C * F)))) * (2.0 * C))) / (Math.pow(B_m, 2.0) - (C * (A * 4.0)));
	} else if (F <= 7.6e+22) {
		tmp = (Math.sqrt(2.0) / B_m) * -Math.sqrt((F * (A + Math.hypot(B_m, A))));
	} else {
		tmp = Math.sqrt((F / B_m)) * -Math.sqrt(2.0);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if F <= -5e-310:
		tmp = -math.sqrt(((2.0 * (-4.0 * (A * (C * F)))) * (2.0 * C))) / (math.pow(B_m, 2.0) - (C * (A * 4.0)))
	elif F <= 7.6e+22:
		tmp = (math.sqrt(2.0) / B_m) * -math.sqrt((F * (A + math.hypot(B_m, A))))
	else:
		tmp = math.sqrt((F / B_m)) * -math.sqrt(2.0)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= -5e-310)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(-4.0 * Float64(A * Float64(C * F)))) * Float64(2.0 * C)))) / Float64((B_m ^ 2.0) - Float64(C * Float64(A * 4.0))));
	elseif (F <= 7.6e+22)
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(-sqrt(Float64(F * Float64(A + hypot(B_m, A))))));
	else
		tmp = Float64(sqrt(Float64(F / B_m)) * Float64(-sqrt(2.0)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= -5e-310)
		tmp = -sqrt(((2.0 * (-4.0 * (A * (C * F)))) * (2.0 * C))) / ((B_m ^ 2.0) - (C * (A * 4.0)));
	elseif (F <= 7.6e+22)
		tmp = (sqrt(2.0) / B_m) * -sqrt((F * (A + hypot(B_m, A))));
	else
		tmp = sqrt((F / B_m)) * -sqrt(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, -5e-310], N[((-N[Sqrt[N[(N[(2.0 * N[(-4.0 * N[(A * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7.6e+22], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[N[(F * N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 14: 37.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)\right) \cdot \left(2 \cdot C\right)}}{{B_m}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;F \leq 2.5 \cdot 10^{-32}:\\ \;\;\;\;\frac{\sqrt{2}}{B_m} \cdot \left(-\sqrt{F \cdot \left(B_m + C\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B_m}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F -5e-310)
   (/
    (- (sqrt (* (* 2.0 (* -4.0 (* A (* C F)))) (* 2.0 C))))
    (- (pow B_m 2.0) (* C (* A 4.0))))
   (if (<= F 2.5e-32)
     (* (/ (sqrt 2.0) B_m) (- (sqrt (* F (+ B_m C)))))
     (* (sqrt (/ F B_m)) (- (sqrt 2.0))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= -5e-310) {
		tmp = -sqrt(((2.0 * (-4.0 * (A * (C * F)))) * (2.0 * C))) / (pow(B_m, 2.0) - (C * (A * 4.0)));
	} else if (F <= 2.5e-32) {
		tmp = (sqrt(2.0) / B_m) * -sqrt((F * (B_m + C)));
	} else {
		tmp = sqrt((F / B_m)) * -sqrt(2.0);
	}
	return tmp;
}

Fortran

B_m = abs(B)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (f <= (-5d-310)) then
        tmp = -sqrt(((2.0d0 * ((-4.0d0) * (a * (c * f)))) * (2.0d0 * c))) / ((b_m ** 2.0d0) - (c * (a * 4.0d0)))
    else if (f <= 2.5d-32) then
        tmp = (sqrt(2.0d0) / b_m) * -sqrt((f * (b_m + c)))
    else
        tmp = sqrt((f / b_m)) * -sqrt(2.0d0)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= -5e-310) {
		tmp = -Math.sqrt(((2.0 * (-4.0 * (A * (C * F)))) * (2.0 * C))) / (Math.pow(B_m, 2.0) - (C * (A * 4.0)));
	} else if (F <= 2.5e-32) {
		tmp = (Math.sqrt(2.0) / B_m) * -Math.sqrt((F * (B_m + C)));
	} else {
		tmp = Math.sqrt((F / B_m)) * -Math.sqrt(2.0);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if F <= -5e-310:
		tmp = -math.sqrt(((2.0 * (-4.0 * (A * (C * F)))) * (2.0 * C))) / (math.pow(B_m, 2.0) - (C * (A * 4.0)))
	elif F <= 2.5e-32:
		tmp = (math.sqrt(2.0) / B_m) * -math.sqrt((F * (B_m + C)))
	else:
		tmp = math.sqrt((F / B_m)) * -math.sqrt(2.0)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= -5e-310)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(-4.0 * Float64(A * Float64(C * F)))) * Float64(2.0 * C)))) / Float64((B_m ^ 2.0) - Float64(C * Float64(A * 4.0))));
	elseif (F <= 2.5e-32)
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(-sqrt(Float64(F * Float64(B_m + C)))));
	else
		tmp = Float64(sqrt(Float64(F / B_m)) * Float64(-sqrt(2.0)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= -5e-310)
		tmp = -sqrt(((2.0 * (-4.0 * (A * (C * F)))) * (2.0 * C))) / ((B_m ^ 2.0) - (C * (A * 4.0)));
	elseif (F <= 2.5e-32)
		tmp = (sqrt(2.0) / B_m) * -sqrt((F * (B_m + C)));
	else
		tmp = sqrt((F / B_m)) * -sqrt(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, -5e-310], N[((-N[Sqrt[N[(N[(2.0 * N[(-4.0 * N[(A * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.5e-32], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[N[(F * N[(B$95$m + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 15: 34.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;F \leq 9.8 \cdot 10^{-32}:\\ \;\;\;\;\frac{\sqrt{2}}{B_m} \cdot \left(-\sqrt{F \cdot \left(B_m + C\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B_m}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F 9.8e-32)
   (* (/ (sqrt 2.0) B_m) (- (sqrt (* F (+ B_m C)))))
   (* (sqrt (/ F B_m)) (- (sqrt 2.0)))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 9.8e-32) {
		tmp = (sqrt(2.0) / B_m) * -sqrt((F * (B_m + C)));
	} else {
		tmp = sqrt((F / B_m)) * -sqrt(2.0);
	}
	return tmp;
}

Fortran

B_m = abs(B)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (f <= 9.8d-32) then
        tmp = (sqrt(2.0d0) / b_m) * -sqrt((f * (b_m + c)))
    else
        tmp = sqrt((f / b_m)) * -sqrt(2.0d0)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 9.8e-32) {
		tmp = (Math.sqrt(2.0) / B_m) * -Math.sqrt((F * (B_m + C)));
	} else {
		tmp = Math.sqrt((F / B_m)) * -Math.sqrt(2.0);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if F <= 9.8e-32:
		tmp = (math.sqrt(2.0) / B_m) * -math.sqrt((F * (B_m + C)))
	else:
		tmp = math.sqrt((F / B_m)) * -math.sqrt(2.0)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= 9.8e-32)
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(-sqrt(Float64(F * Float64(B_m + C)))));
	else
		tmp = Float64(sqrt(Float64(F / B_m)) * Float64(-sqrt(2.0)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= 9.8e-32)
		tmp = (sqrt(2.0) / B_m) * -sqrt((F * (B_m + C)));
	else
		tmp = sqrt((F / B_m)) * -sqrt(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, 9.8e-32], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[N[(F * N[(B$95$m + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 16: 34.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := -\sqrt{2}\\ \mathbf{if}\;F \leq 8.5 \cdot 10^{-32}:\\ \;\;\;\;\frac{t_0}{B_m} \cdot \sqrt{B_m \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B_m}} \cdot t_0\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (sqrt 2.0))))
   (if (<= F 8.5e-32)
     (* (/ t_0 B_m) (sqrt (* B_m F)))
     (* (sqrt (/ F B_m)) t_0))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = -sqrt(2.0);
	double tmp;
	if (F <= 8.5e-32) {
		tmp = (t_0 / B_m) * sqrt((B_m * F));
	} else {
		tmp = sqrt((F / B_m)) * t_0;
	}
	return tmp;
}

Fortran

B_m = abs(B)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -sqrt(2.0d0)
    if (f <= 8.5d-32) then
        tmp = (t_0 / b_m) * sqrt((b_m * f))
    else
        tmp = sqrt((f / b_m)) * t_0
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double t_0 = -Math.sqrt(2.0);
	double tmp;
	if (F <= 8.5e-32) {
		tmp = (t_0 / B_m) * Math.sqrt((B_m * F));
	} else {
		tmp = Math.sqrt((F / B_m)) * t_0;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = -math.sqrt(2.0)
	tmp = 0
	if F <= 8.5e-32:
		tmp = (t_0 / B_m) * math.sqrt((B_m * F))
	else:
		tmp = math.sqrt((F / B_m)) * t_0
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(-sqrt(2.0))
	tmp = 0.0
	if (F <= 8.5e-32)
		tmp = Float64(Float64(t_0 / B_m) * sqrt(Float64(B_m * F)));
	else
		tmp = Float64(sqrt(Float64(F / B_m)) * t_0);
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = -sqrt(2.0);
	tmp = 0.0;
	if (F <= 8.5e-32)
		tmp = (t_0 / B_m) * sqrt((B_m * F));
	else
		tmp = sqrt((F / B_m)) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = (-N[Sqrt[2.0], $MachinePrecision])}, If[LessEqual[F, 8.5e-32], N[(N[(t$95$0 / B$95$m), $MachinePrecision] * N[Sqrt[N[(B$95$m * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 17: 27.4% accurate, 3.1× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \sqrt{\frac{F}{B_m}} \cdot \left(-\sqrt{2}\right) \end{array} \]

FPCore

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

C

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

Fortran

B_m = abs(B)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = sqrt((f / b_m)) * -sqrt(2.0d0)
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	return Math.sqrt((F / B_m)) * -Math.sqrt(2.0);
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	return math.sqrt((F / B_m)) * -math.sqrt(2.0)

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	return Float64(sqrt(Float64(F / B_m)) * Float64(-sqrt(2.0)))
end

MATLAB

B_m = abs(B);
function tmp = code(A, B_m, C, F)
	tmp = sqrt((F / B_m)) * -sqrt(2.0);
end

Wolfram

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

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 18: 5.3% accurate, 5.8× speedup

Math

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

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (* -2.0 (* (/ 1.0 B_m) (sqrt (* C F)))))

C

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

Fortran

B_m = abs(B)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = (-2.0d0) * ((1.0d0 / b_m) * sqrt((c * f)))
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	return -2.0 * ((1.0 / B_m) * Math.sqrt((C * F)));
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	return -2.0 * ((1.0 / B_m) * math.sqrt((C * F)))

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	return Float64(-2.0 * Float64(Float64(1.0 / B_m) * sqrt(Float64(C * F))))
end

MATLAB

B_m = abs(B);
function tmp = code(A, B_m, C, F)
	tmp = -2.0 * ((1.0 / B_m) * sqrt((C * F)));
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := N[(-2.0 * N[(N[(1.0 / B$95$m), $MachinePrecision] * N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Reproduce

herbie shell --seed 2023364 
(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))))