ABCF->ab-angle a

Percentage Accurate: 19.2% → 61.1%

Time: 43.3s

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: 19.2% 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: 61.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := -\sqrt{F}\\ t_1 := {B_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_2 := \sqrt{C + \left(A + \mathsf{hypot}\left(B_m, A - C\right)\right)}\\ t_3 := \frac{-\sqrt{\left(2 \cdot \left(t_1 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t_1}\\ t_4 := \mathsf{fma}\left(A, C \cdot -4, {B_m}^{2}\right)\\ \mathbf{if}\;t_3 \leq -1 \cdot 10^{-195}:\\ \;\;\;\;\frac{t_2 \cdot \left(\sqrt{2 \cdot t_4} \cdot t_0\right)}{\mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;t_3 \leq 10^{-143}:\\ \;\;\;\;\frac{-\sqrt{t_4 \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-0.5, \frac{{B_m}^{2}}{A}, 2 \cdot C\right)\right)}}{t_1}\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;-t_2 \cdot \frac{\sqrt{t_4 \cdot \left(2 \cdot F\right)}}{t_4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B_m} \cdot \left(\sqrt{B_m + C} \cdot t_0\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (sqrt F)))
        (t_1 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_2 (sqrt (+ C (+ A (hypot B_m (- A C))))))
        (t_3
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_1 F))
             (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0)))))))
          t_1))
        (t_4 (fma A (* C -4.0) (pow B_m 2.0))))
   (if (<= t_3 -1e-195)
     (/ (* t_2 (* (sqrt (* 2.0 t_4)) t_0)) (fma B_m B_m (* A (* C -4.0))))
     (if (<= t_3 1e-143)
       (/
        (-
         (sqrt (* t_4 (* (* 2.0 F) (fma -0.5 (/ (pow B_m 2.0) A) (* 2.0 C))))))
        t_1)
       (if (<= t_3 INFINITY)
         (- (* t_2 (/ (sqrt (* t_4 (* 2.0 F))) t_4)))
         (* (/ (sqrt 2.0) B_m) (* (sqrt (+ B_m C)) t_0)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = -sqrt(F);
	double t_1 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_2 = sqrt((C + (A + hypot(B_m, (A - C)))));
	double t_3 = -sqrt(((2.0 * (t_1 * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / t_1;
	double t_4 = fma(A, (C * -4.0), pow(B_m, 2.0));
	double tmp;
	if (t_3 <= -1e-195) {
		tmp = (t_2 * (sqrt((2.0 * t_4)) * t_0)) / fma(B_m, B_m, (A * (C * -4.0)));
	} else if (t_3 <= 1e-143) {
		tmp = -sqrt((t_4 * ((2.0 * F) * fma(-0.5, (pow(B_m, 2.0) / A), (2.0 * C))))) / t_1;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = -(t_2 * (sqrt((t_4 * (2.0 * F))) / t_4));
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt((B_m + C)) * t_0);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(-sqrt(F))
	t_1 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_2 = sqrt(Float64(C + Float64(A + hypot(B_m, Float64(A - C)))))
	t_3 = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_1 * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0))))))) / t_1)
	t_4 = fma(A, Float64(C * -4.0), (B_m ^ 2.0))
	tmp = 0.0
	if (t_3 <= -1e-195)
		tmp = Float64(Float64(t_2 * Float64(sqrt(Float64(2.0 * t_4)) * t_0)) / fma(B_m, B_m, Float64(A * Float64(C * -4.0))));
	elseif (t_3 <= 1e-143)
		tmp = Float64(Float64(-sqrt(Float64(t_4 * Float64(Float64(2.0 * F) * fma(-0.5, Float64((B_m ^ 2.0) / A), Float64(2.0 * C)))))) / t_1);
	elseif (t_3 <= Inf)
		tmp = Float64(-Float64(t_2 * Float64(sqrt(Float64(t_4 * Float64(2.0 * F))) / t_4)));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(Float64(B_m + C)) * t_0));
	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[Sqrt[F], $MachinePrecision])}, Block[{t$95$1 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(C + N[(A + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$1 * 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]) / t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[(A * N[(C * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -1e-195], N[(N[(t$95$2 * N[(N[Sqrt[N[(2.0 * t$95$4), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e-143], N[((-N[Sqrt[N[(t$95$4 * N[(N[(2.0 * F), $MachinePrecision] * N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / A), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], If[LessEqual[t$95$3, Infinity], (-N[(t$95$2 * N[(N[Sqrt[N[(t$95$4 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision]), N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

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

Alternative 2: 61.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, {B_m}^{2}\right)\\ t_1 := C + \left(A + \mathsf{hypot}\left(B_m, A - C\right)\right)\\ t_2 := \mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)\\ t_3 := {B_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_4 := \frac{-\sqrt{\left(2 \cdot \left(t_3 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t_3}\\ \mathbf{if}\;t_4 \leq -4 \cdot 10^{+252}:\\ \;\;\;\;\frac{\left(\sqrt{2 \cdot t_0} \cdot \sqrt{F}\right) \cdot \left(-\sqrt{C + C}\right)}{t_2}\\ \mathbf{elif}\;t_4 \leq -1 \cdot 10^{-195}:\\ \;\;\;\;-\frac{\sqrt{\left(F \cdot t_2\right) \cdot \left(2 \cdot t_1\right)}}{t_2}\\ \mathbf{elif}\;t_4 \leq 10^{-143}:\\ \;\;\;\;\frac{-\sqrt{t_0 \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-0.5, \frac{{B_m}^{2}}{A}, 2 \cdot C\right)\right)}}{t_3}\\ \mathbf{elif}\;t_4 \leq \infty:\\ \;\;\;\;-\sqrt{t_1} \cdot \frac{\sqrt{t_0 \cdot \left(2 \cdot F\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B_m} \cdot \left(\sqrt{B_m + C} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
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 A (* C -4.0) (pow B_m 2.0)))
        (t_1 (+ C (+ A (hypot B_m (- A C)))))
        (t_2 (fma B_m B_m (* A (* C -4.0))))
        (t_3 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_4
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_3 F))
             (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0)))))))
          t_3)))
   (if (<= t_4 -4e+252)
     (/ (* (* (sqrt (* 2.0 t_0)) (sqrt F)) (- (sqrt (+ C C)))) t_2)
     (if (<= t_4 -1e-195)
       (- (/ (sqrt (* (* F t_2) (* 2.0 t_1))) t_2))
       (if (<= t_4 1e-143)
         (/
          (-
           (sqrt
            (* t_0 (* (* 2.0 F) (fma -0.5 (/ (pow B_m 2.0) A) (* 2.0 C))))))
          t_3)
         (if (<= t_4 INFINITY)
           (- (* (sqrt t_1) (/ (sqrt (* t_0 (* 2.0 F))) t_0)))
           (* (/ (sqrt 2.0) B_m) (* (sqrt (+ B_m C)) (- (sqrt F))))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(A, (C * -4.0), pow(B_m, 2.0));
	double t_1 = C + (A + hypot(B_m, (A - C)));
	double t_2 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_3 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_4 = -sqrt(((2.0 * (t_3 * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / t_3;
	double tmp;
	if (t_4 <= -4e+252) {
		tmp = ((sqrt((2.0 * t_0)) * sqrt(F)) * -sqrt((C + C))) / t_2;
	} else if (t_4 <= -1e-195) {
		tmp = -(sqrt(((F * t_2) * (2.0 * t_1))) / t_2);
	} else if (t_4 <= 1e-143) {
		tmp = -sqrt((t_0 * ((2.0 * F) * fma(-0.5, (pow(B_m, 2.0) / A), (2.0 * C))))) / t_3;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = -(sqrt(t_1) * (sqrt((t_0 * (2.0 * F))) / t_0));
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt((B_m + C)) * -sqrt(F));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(A, Float64(C * -4.0), (B_m ^ 2.0))
	t_1 = Float64(C + Float64(A + hypot(B_m, Float64(A - C))))
	t_2 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_3 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_4 = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_3 * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0))))))) / t_3)
	tmp = 0.0
	if (t_4 <= -4e+252)
		tmp = Float64(Float64(Float64(sqrt(Float64(2.0 * t_0)) * sqrt(F)) * Float64(-sqrt(Float64(C + C)))) / t_2);
	elseif (t_4 <= -1e-195)
		tmp = Float64(-Float64(sqrt(Float64(Float64(F * t_2) * Float64(2.0 * t_1))) / t_2));
	elseif (t_4 <= 1e-143)
		tmp = Float64(Float64(-sqrt(Float64(t_0 * Float64(Float64(2.0 * F) * fma(-0.5, Float64((B_m ^ 2.0) / A), Float64(2.0 * C)))))) / t_3);
	elseif (t_4 <= Inf)
		tmp = Float64(-Float64(sqrt(t_1) * Float64(sqrt(Float64(t_0 * Float64(2.0 * F))) / t_0)));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(Float64(B_m + C)) * Float64(-sqrt(F))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
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[(A * N[(C * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(C + N[(A + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$3 * 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]) / t$95$3), $MachinePrecision]}, If[LessEqual[t$95$4, -4e+252], N[(N[(N[(N[Sqrt[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * (-N[Sqrt[N[(C + C), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$4, -1e-195], (-N[(N[Sqrt[N[(N[(F * t$95$2), $MachinePrecision] * N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision]), If[LessEqual[t$95$4, 1e-143], N[((-N[Sqrt[N[(t$95$0 * N[(N[(2.0 * F), $MachinePrecision] * N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / A), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$3), $MachinePrecision], If[LessEqual[t$95$4, Infinity], (-N[(N[Sqrt[t$95$1], $MachinePrecision] * N[(N[Sqrt[N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

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

Alternative 3: 61.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, {B_m}^{2}\right)\\ t_1 := \mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)\\ t_2 := {B_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{-\sqrt{\left(2 \cdot \left(t_2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t_2}\\ t_4 := -\sqrt{C + C}\\ \mathbf{if}\;t_3 \leq -4 \cdot 10^{+252}:\\ \;\;\;\;\frac{\left(\sqrt{2 \cdot t_0} \cdot \sqrt{F}\right) \cdot t_4}{t_1}\\ \mathbf{elif}\;t_3 \leq -1 \cdot 10^{-195}:\\ \;\;\;\;-\frac{\sqrt{\left(F \cdot t_1\right) \cdot \left(2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B_m, A - C\right)\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;t_3 \leq 10^{-143}:\\ \;\;\;\;\frac{-\sqrt{t_0 \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-0.5, \frac{{B_m}^{2}}{A}, 2 \cdot C\right)\right)}}{t_2}\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left({B_m}^{2} + -4 \cdot \left(A \cdot C\right)\right)\right)} \cdot t_4}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B_m} \cdot \left(\sqrt{B_m + C} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
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 A (* C -4.0) (pow B_m 2.0)))
        (t_1 (fma B_m B_m (* A (* C -4.0))))
        (t_2 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_3
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_2 F))
             (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0)))))))
          t_2))
        (t_4 (- (sqrt (+ C C)))))
   (if (<= t_3 -4e+252)
     (/ (* (* (sqrt (* 2.0 t_0)) (sqrt F)) t_4) t_1)
     (if (<= t_3 -1e-195)
       (- (/ (sqrt (* (* F t_1) (* 2.0 (+ C (+ A (hypot B_m (- A C))))))) t_1))
       (if (<= t_3 1e-143)
         (/
          (-
           (sqrt
            (* t_0 (* (* 2.0 F) (fma -0.5 (/ (pow B_m 2.0) A) (* 2.0 C))))))
          t_2)
         (if (<= t_3 INFINITY)
           (/
            (* (sqrt (* 2.0 (* F (+ (pow B_m 2.0) (* -4.0 (* A C)))))) t_4)
            t_1)
           (* (/ (sqrt 2.0) B_m) (* (sqrt (+ B_m C)) (- (sqrt F))))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(A, (C * -4.0), pow(B_m, 2.0));
	double t_1 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_2 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_3 = -sqrt(((2.0 * (t_2 * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / t_2;
	double t_4 = -sqrt((C + C));
	double tmp;
	if (t_3 <= -4e+252) {
		tmp = ((sqrt((2.0 * t_0)) * sqrt(F)) * t_4) / t_1;
	} else if (t_3 <= -1e-195) {
		tmp = -(sqrt(((F * t_1) * (2.0 * (C + (A + hypot(B_m, (A - C))))))) / t_1);
	} else if (t_3 <= 1e-143) {
		tmp = -sqrt((t_0 * ((2.0 * F) * fma(-0.5, (pow(B_m, 2.0) / A), (2.0 * C))))) / t_2;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = (sqrt((2.0 * (F * (pow(B_m, 2.0) + (-4.0 * (A * C)))))) * t_4) / t_1;
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt((B_m + C)) * -sqrt(F));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(A, Float64(C * -4.0), (B_m ^ 2.0))
	t_1 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_2 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_3 = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_2 * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0))))))) / t_2)
	t_4 = Float64(-sqrt(Float64(C + C)))
	tmp = 0.0
	if (t_3 <= -4e+252)
		tmp = Float64(Float64(Float64(sqrt(Float64(2.0 * t_0)) * sqrt(F)) * t_4) / t_1);
	elseif (t_3 <= -1e-195)
		tmp = Float64(-Float64(sqrt(Float64(Float64(F * t_1) * Float64(2.0 * Float64(C + Float64(A + hypot(B_m, Float64(A - C))))))) / t_1));
	elseif (t_3 <= 1e-143)
		tmp = Float64(Float64(-sqrt(Float64(t_0 * Float64(Float64(2.0 * F) * fma(-0.5, Float64((B_m ^ 2.0) / A), Float64(2.0 * C)))))) / t_2);
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(F * Float64((B_m ^ 2.0) + Float64(-4.0 * Float64(A * C)))))) * t_4) / t_1);
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(Float64(B_m + C)) * Float64(-sqrt(F))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
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[(A * N[(C * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $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[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$2 * 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]) / t$95$2), $MachinePrecision]}, Block[{t$95$4 = (-N[Sqrt[N[(C + C), $MachinePrecision]], $MachinePrecision])}, If[LessEqual[t$95$3, -4e+252], N[(N[(N[(N[Sqrt[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$3, -1e-195], (-N[(N[Sqrt[N[(N[(F * t$95$1), $MachinePrecision] * N[(2.0 * N[(C + N[(A + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision]), If[LessEqual[t$95$3, 1e-143], N[((-N[Sqrt[N[(t$95$0 * N[(N[(2.0 * F), $MachinePrecision] * N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / A), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[(N[Sqrt[N[(2.0 * N[(F * N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$4), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

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

Alternative 4: 54.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := {B_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_1 := \mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)\\ t_2 := \frac{-\sqrt{\left(2 \cdot \left(t_0 \cdot F\right)\right) \cdot \left(2 \cdot C + -0.5 \cdot \left(B_m \cdot \left(B_m \cdot \frac{1}{A}\right)\right)\right)}}{t_0}\\ t_3 := C + \left(A + \mathsf{hypot}\left(B_m, A - C\right)\right)\\ t_4 := -\sqrt{F}\\ \mathbf{if}\;{B_m}^{2} \leq 2 \cdot 10^{-17}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;{B_m}^{2} \leq 1.55 \cdot 10^{+60}:\\ \;\;\;\;-\frac{\sqrt{\left(F \cdot t_1\right) \cdot \left(2 \cdot t_3\right)}}{t_1}\\ \mathbf{elif}\;{B_m}^{2} \leq 5 \cdot 10^{+92}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;{B_m}^{2} \leq 10^{+239}:\\ \;\;\;\;\frac{\sqrt{t_3} \cdot \left(B_m \cdot \left(\sqrt{2} \cdot t_4\right)\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B_m} \cdot \left(\sqrt{B_m + C} \cdot t_4\right)\\ \end{array} \end{array} \]

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 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_1 (fma B_m B_m (* A (* C -4.0))))
        (t_2
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_0 F))
             (+ (* 2.0 C) (* -0.5 (* B_m (* B_m (/ 1.0 A))))))))
          t_0))
        (t_3 (+ C (+ A (hypot B_m (- A C)))))
        (t_4 (- (sqrt F))))
   (if (<= (pow B_m 2.0) 2e-17)
     t_2
     (if (<= (pow B_m 2.0) 1.55e+60)
       (- (/ (sqrt (* (* F t_1) (* 2.0 t_3))) t_1))
       (if (<= (pow B_m 2.0) 5e+92)
         t_2
         (if (<= (pow B_m 2.0) 1e+239)
           (/ (* (sqrt t_3) (* B_m (* (sqrt 2.0) t_4))) t_1)
           (* (/ (sqrt 2.0) B_m) (* (sqrt (+ B_m C)) t_4))))))))

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 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_1 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_2 = -sqrt(((2.0 * (t_0 * F)) * ((2.0 * C) + (-0.5 * (B_m * (B_m * (1.0 / A))))))) / t_0;
	double t_3 = C + (A + hypot(B_m, (A - C)));
	double t_4 = -sqrt(F);
	double tmp;
	if (pow(B_m, 2.0) <= 2e-17) {
		tmp = t_2;
	} else if (pow(B_m, 2.0) <= 1.55e+60) {
		tmp = -(sqrt(((F * t_1) * (2.0 * t_3))) / t_1);
	} else if (pow(B_m, 2.0) <= 5e+92) {
		tmp = t_2;
	} else if (pow(B_m, 2.0) <= 1e+239) {
		tmp = (sqrt(t_3) * (B_m * (sqrt(2.0) * t_4))) / t_1;
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt((B_m + C)) * t_4);
	}
	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((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_1 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_2 = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_0 * F)) * Float64(Float64(2.0 * C) + Float64(-0.5 * Float64(B_m * Float64(B_m * Float64(1.0 / A)))))))) / t_0)
	t_3 = Float64(C + Float64(A + hypot(B_m, Float64(A - C))))
	t_4 = Float64(-sqrt(F))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-17)
		tmp = t_2;
	elseif ((B_m ^ 2.0) <= 1.55e+60)
		tmp = Float64(-Float64(sqrt(Float64(Float64(F * t_1) * Float64(2.0 * t_3))) / t_1));
	elseif ((B_m ^ 2.0) <= 5e+92)
		tmp = t_2;
	elseif ((B_m ^ 2.0) <= 1e+239)
		tmp = Float64(Float64(sqrt(t_3) * Float64(B_m * Float64(sqrt(2.0) * t_4))) / t_1);
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(Float64(B_m + C)) * t_4));
	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[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $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[N[(N[(2.0 * N[(t$95$0 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * C), $MachinePrecision] + N[(-0.5 * N[(B$95$m * N[(B$95$m * N[(1.0 / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision]}, Block[{t$95$3 = N[(C + N[(A + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = (-N[Sqrt[F], $MachinePrecision])}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-17], t$95$2, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1.55e+60], (-N[(N[Sqrt[N[(N[(F * t$95$1), $MachinePrecision] * N[(2.0 * t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision]), If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+92], t$95$2, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+239], N[(N[(N[Sqrt[t$95$3], $MachinePrecision] * N[(B$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

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

Alternative 5: 54.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := {B_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_1 := \frac{-\sqrt{\left(2 \cdot \left(t_0 \cdot F\right)\right) \cdot \left(2 \cdot C + -0.5 \cdot \left(B_m \cdot \left(B_m \cdot \frac{1}{A}\right)\right)\right)}}{t_0}\\ t_2 := C + \left(A + \mathsf{hypot}\left(B_m, A - C\right)\right)\\ t_3 := -\sqrt{F}\\ t_4 := \mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B_m}^{2} \leq 2 \cdot 10^{-17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;{B_m}^{2} \leq 2 \cdot 10^{+60}:\\ \;\;\;\;\frac{-\sqrt{F \cdot \left(\mathsf{fma}\left(A, C \cdot -4, {B_m}^{2}\right) \cdot \left(2 \cdot t_2\right)\right)}}{t_4}\\ \mathbf{elif}\;{B_m}^{2} \leq 5 \cdot 10^{+92}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;{B_m}^{2} \leq 10^{+239}:\\ \;\;\;\;\frac{\sqrt{t_2} \cdot \left(B_m \cdot \left(\sqrt{2} \cdot t_3\right)\right)}{t_4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B_m} \cdot \left(\sqrt{B_m + C} \cdot t_3\right)\\ \end{array} \end{array} \]

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 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_1
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_0 F))
             (+ (* 2.0 C) (* -0.5 (* B_m (* B_m (/ 1.0 A))))))))
          t_0))
        (t_2 (+ C (+ A (hypot B_m (- A C)))))
        (t_3 (- (sqrt F)))
        (t_4 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= (pow B_m 2.0) 2e-17)
     t_1
     (if (<= (pow B_m 2.0) 2e+60)
       (/
        (- (sqrt (* F (* (fma A (* C -4.0) (pow B_m 2.0)) (* 2.0 t_2)))))
        t_4)
       (if (<= (pow B_m 2.0) 5e+92)
         t_1
         (if (<= (pow B_m 2.0) 1e+239)
           (/ (* (sqrt t_2) (* B_m (* (sqrt 2.0) t_3))) t_4)
           (* (/ (sqrt 2.0) B_m) (* (sqrt (+ B_m C)) t_3))))))))

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 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_1 = -sqrt(((2.0 * (t_0 * F)) * ((2.0 * C) + (-0.5 * (B_m * (B_m * (1.0 / A))))))) / t_0;
	double t_2 = C + (A + hypot(B_m, (A - C)));
	double t_3 = -sqrt(F);
	double t_4 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (pow(B_m, 2.0) <= 2e-17) {
		tmp = t_1;
	} else if (pow(B_m, 2.0) <= 2e+60) {
		tmp = -sqrt((F * (fma(A, (C * -4.0), pow(B_m, 2.0)) * (2.0 * t_2)))) / t_4;
	} else if (pow(B_m, 2.0) <= 5e+92) {
		tmp = t_1;
	} else if (pow(B_m, 2.0) <= 1e+239) {
		tmp = (sqrt(t_2) * (B_m * (sqrt(2.0) * t_3))) / t_4;
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt((B_m + C)) * t_3);
	}
	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((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_1 = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_0 * F)) * Float64(Float64(2.0 * C) + Float64(-0.5 * Float64(B_m * Float64(B_m * Float64(1.0 / A)))))))) / t_0)
	t_2 = Float64(C + Float64(A + hypot(B_m, Float64(A - C))))
	t_3 = Float64(-sqrt(F))
	t_4 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-17)
		tmp = t_1;
	elseif ((B_m ^ 2.0) <= 2e+60)
		tmp = Float64(Float64(-sqrt(Float64(F * Float64(fma(A, Float64(C * -4.0), (B_m ^ 2.0)) * Float64(2.0 * t_2))))) / t_4);
	elseif ((B_m ^ 2.0) <= 5e+92)
		tmp = t_1;
	elseif ((B_m ^ 2.0) <= 1e+239)
		tmp = Float64(Float64(sqrt(t_2) * Float64(B_m * Float64(sqrt(2.0) * t_3))) / t_4);
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(Float64(B_m + C)) * t_3));
	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[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$0 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * C), $MachinePrecision] + N[(-0.5 * N[(B$95$m * N[(B$95$m * N[(1.0 / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(C + N[(A + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = (-N[Sqrt[F], $MachinePrecision])}, Block[{t$95$4 = 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], 2e-17], t$95$1, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+60], N[((-N[Sqrt[N[(F * N[(N[(A * N[(C * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(2.0 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$4), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+92], t$95$1, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+239], N[(N[(N[Sqrt[t$95$2], $MachinePrecision] * N[(B$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

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

Alternative 6: 55.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \sqrt{C + \left(A + \mathsf{hypot}\left(B_m, A - C\right)\right)}\\ t_1 := -\sqrt{F}\\ t_2 := {B_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{-\sqrt{\left(2 \cdot \left(t_2 \cdot F\right)\right) \cdot \left(2 \cdot C + -0.5 \cdot \left(B_m \cdot \left(B_m \cdot \frac{1}{A}\right)\right)\right)}}{t_2}\\ t_4 := \mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B_m}^{2} \leq 2 \cdot 10^{-124}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;{B_m}^{2} \leq 2 \cdot 10^{+63}:\\ \;\;\;\;\frac{t_0 \cdot \left(-\sqrt{F \cdot \left(2 \cdot \mathsf{fma}\left(A, C \cdot -4, {B_m}^{2}\right)\right)}\right)}{t_4}\\ \mathbf{elif}\;{B_m}^{2} \leq 5 \cdot 10^{+92}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;{B_m}^{2} \leq 10^{+239}:\\ \;\;\;\;\frac{t_0 \cdot \left(B_m \cdot \left(\sqrt{2} \cdot t_1\right)\right)}{t_4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B_m} \cdot \left(\sqrt{B_m + C} \cdot t_1\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (sqrt (+ C (+ A (hypot B_m (- A C))))))
        (t_1 (- (sqrt F)))
        (t_2 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_3
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_2 F))
             (+ (* 2.0 C) (* -0.5 (* B_m (* B_m (/ 1.0 A))))))))
          t_2))
        (t_4 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= (pow B_m 2.0) 2e-124)
     t_3
     (if (<= (pow B_m 2.0) 2e+63)
       (/
        (* t_0 (- (sqrt (* F (* 2.0 (fma A (* C -4.0) (pow B_m 2.0)))))))
        t_4)
       (if (<= (pow B_m 2.0) 5e+92)
         t_3
         (if (<= (pow B_m 2.0) 1e+239)
           (/ (* t_0 (* B_m (* (sqrt 2.0) t_1))) t_4)
           (* (/ (sqrt 2.0) B_m) (* (sqrt (+ B_m C)) t_1))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = sqrt((C + (A + hypot(B_m, (A - C)))));
	double t_1 = -sqrt(F);
	double t_2 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_3 = -sqrt(((2.0 * (t_2 * F)) * ((2.0 * C) + (-0.5 * (B_m * (B_m * (1.0 / A))))))) / t_2;
	double t_4 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (pow(B_m, 2.0) <= 2e-124) {
		tmp = t_3;
	} else if (pow(B_m, 2.0) <= 2e+63) {
		tmp = (t_0 * -sqrt((F * (2.0 * fma(A, (C * -4.0), pow(B_m, 2.0)))))) / t_4;
	} else if (pow(B_m, 2.0) <= 5e+92) {
		tmp = t_3;
	} else if (pow(B_m, 2.0) <= 1e+239) {
		tmp = (t_0 * (B_m * (sqrt(2.0) * t_1))) / t_4;
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt((B_m + C)) * t_1);
	}
	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 = sqrt(Float64(C + Float64(A + hypot(B_m, Float64(A - C)))))
	t_1 = Float64(-sqrt(F))
	t_2 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_3 = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_2 * F)) * Float64(Float64(2.0 * C) + Float64(-0.5 * Float64(B_m * Float64(B_m * Float64(1.0 / A)))))))) / t_2)
	t_4 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-124)
		tmp = t_3;
	elseif ((B_m ^ 2.0) <= 2e+63)
		tmp = Float64(Float64(t_0 * Float64(-sqrt(Float64(F * Float64(2.0 * fma(A, Float64(C * -4.0), (B_m ^ 2.0))))))) / t_4);
	elseif ((B_m ^ 2.0) <= 5e+92)
		tmp = t_3;
	elseif ((B_m ^ 2.0) <= 1e+239)
		tmp = Float64(Float64(t_0 * Float64(B_m * Float64(sqrt(2.0) * t_1))) / t_4);
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(Float64(B_m + C)) * t_1));
	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[Sqrt[N[(C + N[(A + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = (-N[Sqrt[F], $MachinePrecision])}, Block[{t$95$2 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$2 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * C), $MachinePrecision] + N[(-0.5 * N[(B$95$m * N[(B$95$m * N[(1.0 / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision]}, Block[{t$95$4 = 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], 2e-124], t$95$3, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+63], N[(N[(t$95$0 * (-N[Sqrt[N[(F * N[(2.0 * N[(A * N[(C * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+92], t$95$3, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+239], N[(N[(t$95$0 * N[(B$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

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

Alternative 7: 55.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := C + \left(A + \mathsf{hypot}\left(B_m, A - C\right)\right)\\ t_1 := -\sqrt{F}\\ t_2 := {B_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{-\sqrt{\left(2 \cdot \left(t_2 \cdot F\right)\right) \cdot \left(2 \cdot C + -0.5 \cdot \left(B_m \cdot \left(B_m \cdot \frac{1}{A}\right)\right)\right)}}{t_2}\\ t_4 := \mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B_m}^{2} \leq 2 \cdot 10^{-124}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;{B_m}^{2} \leq 2 \cdot 10^{+63}:\\ \;\;\;\;\frac{\sqrt{F \cdot \mathsf{fma}\left(A, C \cdot -4, {B_m}^{2}\right)} \cdot \left(-\sqrt{2 \cdot t_0}\right)}{t_4}\\ \mathbf{elif}\;{B_m}^{2} \leq 5 \cdot 10^{+92}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;{B_m}^{2} \leq 10^{+239}:\\ \;\;\;\;\frac{\sqrt{t_0} \cdot \left(B_m \cdot \left(\sqrt{2} \cdot t_1\right)\right)}{t_4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B_m} \cdot \left(\sqrt{B_m + C} \cdot t_1\right)\\ \end{array} \end{array} \]

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 (+ C (+ A (hypot B_m (- A C)))))
        (t_1 (- (sqrt F)))
        (t_2 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_3
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_2 F))
             (+ (* 2.0 C) (* -0.5 (* B_m (* B_m (/ 1.0 A))))))))
          t_2))
        (t_4 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= (pow B_m 2.0) 2e-124)
     t_3
     (if (<= (pow B_m 2.0) 2e+63)
       (/
        (*
         (sqrt (* F (fma A (* C -4.0) (pow B_m 2.0))))
         (- (sqrt (* 2.0 t_0))))
        t_4)
       (if (<= (pow B_m 2.0) 5e+92)
         t_3
         (if (<= (pow B_m 2.0) 1e+239)
           (/ (* (sqrt t_0) (* B_m (* (sqrt 2.0) t_1))) t_4)
           (* (/ (sqrt 2.0) B_m) (* (sqrt (+ B_m C)) t_1))))))))

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 = C + (A + hypot(B_m, (A - C)));
	double t_1 = -sqrt(F);
	double t_2 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_3 = -sqrt(((2.0 * (t_2 * F)) * ((2.0 * C) + (-0.5 * (B_m * (B_m * (1.0 / A))))))) / t_2;
	double t_4 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (pow(B_m, 2.0) <= 2e-124) {
		tmp = t_3;
	} else if (pow(B_m, 2.0) <= 2e+63) {
		tmp = (sqrt((F * fma(A, (C * -4.0), pow(B_m, 2.0)))) * -sqrt((2.0 * t_0))) / t_4;
	} else if (pow(B_m, 2.0) <= 5e+92) {
		tmp = t_3;
	} else if (pow(B_m, 2.0) <= 1e+239) {
		tmp = (sqrt(t_0) * (B_m * (sqrt(2.0) * t_1))) / t_4;
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt((B_m + C)) * t_1);
	}
	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(C + Float64(A + hypot(B_m, Float64(A - C))))
	t_1 = Float64(-sqrt(F))
	t_2 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_3 = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_2 * F)) * Float64(Float64(2.0 * C) + Float64(-0.5 * Float64(B_m * Float64(B_m * Float64(1.0 / A)))))))) / t_2)
	t_4 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-124)
		tmp = t_3;
	elseif ((B_m ^ 2.0) <= 2e+63)
		tmp = Float64(Float64(sqrt(Float64(F * fma(A, Float64(C * -4.0), (B_m ^ 2.0)))) * Float64(-sqrt(Float64(2.0 * t_0)))) / t_4);
	elseif ((B_m ^ 2.0) <= 5e+92)
		tmp = t_3;
	elseif ((B_m ^ 2.0) <= 1e+239)
		tmp = Float64(Float64(sqrt(t_0) * Float64(B_m * Float64(sqrt(2.0) * t_1))) / t_4);
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(Float64(B_m + C)) * t_1));
	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[(C + N[(A + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[Sqrt[F], $MachinePrecision])}, Block[{t$95$2 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$2 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * C), $MachinePrecision] + N[(-0.5 * N[(B$95$m * N[(B$95$m * N[(1.0 / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision]}, Block[{t$95$4 = 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], 2e-124], t$95$3, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+63], N[(N[(N[Sqrt[N[(F * N[(A * N[(C * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+92], t$95$3, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+239], N[(N[(N[Sqrt[t$95$0], $MachinePrecision] * N[(B$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

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

Alternative 8: 54.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := C + \left(A + \mathsf{hypot}\left(B_m, A - C\right)\right)\\ t_1 := -\sqrt{F}\\ t_2 := {B_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{-\sqrt{\left(2 \cdot \left(t_2 \cdot F\right)\right) \cdot \left(2 \cdot C + -0.5 \cdot \left(B_m \cdot \left(B_m \cdot \frac{1}{A}\right)\right)\right)}}{t_2}\\ t_4 := \mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B_m}^{2} \leq 2 \cdot 10^{-124}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;{B_m}^{2} \leq 2 \cdot 10^{+60}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(2 \cdot t_0\right)} \cdot \left(-\sqrt{\mathsf{fma}\left(A, C \cdot -4, {B_m}^{2}\right)}\right)}{t_4}\\ \mathbf{elif}\;{B_m}^{2} \leq 5 \cdot 10^{+92}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;{B_m}^{2} \leq 10^{+239}:\\ \;\;\;\;\frac{\sqrt{t_0} \cdot \left(B_m \cdot \left(\sqrt{2} \cdot t_1\right)\right)}{t_4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B_m} \cdot \left(\sqrt{B_m + C} \cdot t_1\right)\\ \end{array} \end{array} \]

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 (+ C (+ A (hypot B_m (- A C)))))
        (t_1 (- (sqrt F)))
        (t_2 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_3
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_2 F))
             (+ (* 2.0 C) (* -0.5 (* B_m (* B_m (/ 1.0 A))))))))
          t_2))
        (t_4 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= (pow B_m 2.0) 2e-124)
     t_3
     (if (<= (pow B_m 2.0) 2e+60)
       (/
        (*
         (sqrt (* F (* 2.0 t_0)))
         (- (sqrt (fma A (* C -4.0) (pow B_m 2.0)))))
        t_4)
       (if (<= (pow B_m 2.0) 5e+92)
         t_3
         (if (<= (pow B_m 2.0) 1e+239)
           (/ (* (sqrt t_0) (* B_m (* (sqrt 2.0) t_1))) t_4)
           (* (/ (sqrt 2.0) B_m) (* (sqrt (+ B_m C)) t_1))))))))

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 = C + (A + hypot(B_m, (A - C)));
	double t_1 = -sqrt(F);
	double t_2 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_3 = -sqrt(((2.0 * (t_2 * F)) * ((2.0 * C) + (-0.5 * (B_m * (B_m * (1.0 / A))))))) / t_2;
	double t_4 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (pow(B_m, 2.0) <= 2e-124) {
		tmp = t_3;
	} else if (pow(B_m, 2.0) <= 2e+60) {
		tmp = (sqrt((F * (2.0 * t_0))) * -sqrt(fma(A, (C * -4.0), pow(B_m, 2.0)))) / t_4;
	} else if (pow(B_m, 2.0) <= 5e+92) {
		tmp = t_3;
	} else if (pow(B_m, 2.0) <= 1e+239) {
		tmp = (sqrt(t_0) * (B_m * (sqrt(2.0) * t_1))) / t_4;
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt((B_m + C)) * t_1);
	}
	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(C + Float64(A + hypot(B_m, Float64(A - C))))
	t_1 = Float64(-sqrt(F))
	t_2 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_3 = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_2 * F)) * Float64(Float64(2.0 * C) + Float64(-0.5 * Float64(B_m * Float64(B_m * Float64(1.0 / A)))))))) / t_2)
	t_4 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-124)
		tmp = t_3;
	elseif ((B_m ^ 2.0) <= 2e+60)
		tmp = Float64(Float64(sqrt(Float64(F * Float64(2.0 * t_0))) * Float64(-sqrt(fma(A, Float64(C * -4.0), (B_m ^ 2.0))))) / t_4);
	elseif ((B_m ^ 2.0) <= 5e+92)
		tmp = t_3;
	elseif ((B_m ^ 2.0) <= 1e+239)
		tmp = Float64(Float64(sqrt(t_0) * Float64(B_m * Float64(sqrt(2.0) * t_1))) / t_4);
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(Float64(B_m + C)) * t_1));
	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[(C + N[(A + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[Sqrt[F], $MachinePrecision])}, Block[{t$95$2 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$2 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * C), $MachinePrecision] + N[(-0.5 * N[(B$95$m * N[(B$95$m * N[(1.0 / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision]}, Block[{t$95$4 = 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], 2e-124], t$95$3, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+60], N[(N[(N[Sqrt[N[(F * N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(A * N[(C * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+92], t$95$3, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+239], N[(N[(N[Sqrt[t$95$0], $MachinePrecision] * N[(B$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

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

Alternative 9: 52.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := {B_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_1 := \mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)\\ t_2 := \frac{-\sqrt{\left(2 \cdot \left(t_0 \cdot F\right)\right) \cdot \left(2 \cdot C + -0.5 \cdot \left(B_m \cdot \left(B_m \cdot \frac{1}{A}\right)\right)\right)}}{t_0}\\ \mathbf{if}\;{B_m}^{2} \leq 2 \cdot 10^{-17}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;{B_m}^{2} \leq 1.55 \cdot 10^{+60}:\\ \;\;\;\;-\frac{\sqrt{\left(F \cdot t_1\right) \cdot \left(2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B_m, A - C\right)\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;{B_m}^{2} \leq 5 \cdot 10^{+92}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B_m} \cdot \left(\sqrt{B_m + C} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
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 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_1 (fma B_m B_m (* A (* C -4.0))))
        (t_2
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_0 F))
             (+ (* 2.0 C) (* -0.5 (* B_m (* B_m (/ 1.0 A))))))))
          t_0)))
   (if (<= (pow B_m 2.0) 2e-17)
     t_2
     (if (<= (pow B_m 2.0) 1.55e+60)
       (- (/ (sqrt (* (* F t_1) (* 2.0 (+ C (+ A (hypot B_m (- A C))))))) t_1))
       (if (<= (pow B_m 2.0) 5e+92)
         t_2
         (* (/ (sqrt 2.0) B_m) (* (sqrt (+ B_m C)) (- (sqrt F)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_1 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_2 = -sqrt(((2.0 * (t_0 * F)) * ((2.0 * C) + (-0.5 * (B_m * (B_m * (1.0 / A))))))) / t_0;
	double tmp;
	if (pow(B_m, 2.0) <= 2e-17) {
		tmp = t_2;
	} else if (pow(B_m, 2.0) <= 1.55e+60) {
		tmp = -(sqrt(((F * t_1) * (2.0 * (C + (A + hypot(B_m, (A - C))))))) / t_1);
	} else if (pow(B_m, 2.0) <= 5e+92) {
		tmp = t_2;
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt((B_m + C)) * -sqrt(F));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_1 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_2 = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_0 * F)) * Float64(Float64(2.0 * C) + Float64(-0.5 * Float64(B_m * Float64(B_m * Float64(1.0 / A)))))))) / t_0)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-17)
		tmp = t_2;
	elseif ((B_m ^ 2.0) <= 1.55e+60)
		tmp = Float64(-Float64(sqrt(Float64(Float64(F * t_1) * Float64(2.0 * Float64(C + Float64(A + hypot(B_m, Float64(A - C))))))) / t_1));
	elseif ((B_m ^ 2.0) <= 5e+92)
		tmp = t_2;
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(Float64(B_m + C)) * Float64(-sqrt(F))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
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[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $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[N[(N[(2.0 * N[(t$95$0 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * C), $MachinePrecision] + N[(-0.5 * N[(B$95$m * N[(B$95$m * N[(1.0 / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-17], t$95$2, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1.55e+60], (-N[(N[Sqrt[N[(N[(F * t$95$1), $MachinePrecision] * N[(2.0 * N[(C + N[(A + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision]), If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+92], t$95$2, N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

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

Alternative 10: 51.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := {B_m}^{2} - \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;{B_m}^{2} \leq 5 \cdot 10^{+92}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(t_0 \cdot F\right)\right) \cdot \left(2 \cdot C + -0.5 \cdot \left(B_m \cdot \left(B_m \cdot \frac{1}{A}\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B_m} \cdot \left(\sqrt{B_m + C} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
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 (- (pow B_m 2.0) (* (* 4.0 A) C))))
   (if (<= (pow B_m 2.0) 5e+92)
     (/
      (-
       (sqrt
        (*
         (* 2.0 (* t_0 F))
         (+ (* 2.0 C) (* -0.5 (* B_m (* B_m (/ 1.0 A))))))))
      t_0)
     (* (/ (sqrt 2.0) B_m) (* (sqrt (+ B_m C)) (- (sqrt F)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double tmp;
	if (pow(B_m, 2.0) <= 5e+92) {
		tmp = -sqrt(((2.0 * (t_0 * F)) * ((2.0 * C) + (-0.5 * (B_m * (B_m * (1.0 / A))))))) / t_0;
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt((B_m + C)) * -sqrt(F));
	}
	return tmp;
}

Fortran

B_m = abs(B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b_m ** 2.0d0) - ((4.0d0 * a) * c)
    if ((b_m ** 2.0d0) <= 5d+92) then
        tmp = -sqrt(((2.0d0 * (t_0 * f)) * ((2.0d0 * c) + ((-0.5d0) * (b_m * (b_m * (1.0d0 / a))))))) / t_0
    else
        tmp = (sqrt(2.0d0) / b_m) * (sqrt((b_m + c)) * -sqrt(f))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = Math.pow(B_m, 2.0) - ((4.0 * A) * C);
	double tmp;
	if (Math.pow(B_m, 2.0) <= 5e+92) {
		tmp = -Math.sqrt(((2.0 * (t_0 * F)) * ((2.0 * C) + (-0.5 * (B_m * (B_m * (1.0 / A))))))) / t_0;
	} else {
		tmp = (Math.sqrt(2.0) / B_m) * (Math.sqrt((B_m + C)) * -Math.sqrt(F));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = math.pow(B_m, 2.0) - ((4.0 * A) * C)
	tmp = 0
	if math.pow(B_m, 2.0) <= 5e+92:
		tmp = -math.sqrt(((2.0 * (t_0 * F)) * ((2.0 * C) + (-0.5 * (B_m * (B_m * (1.0 / A))))))) / t_0
	else:
		tmp = (math.sqrt(2.0) / B_m) * (math.sqrt((B_m + C)) * -math.sqrt(F))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e+92)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_0 * F)) * Float64(Float64(2.0 * C) + Float64(-0.5 * Float64(B_m * Float64(B_m * Float64(1.0 / A)))))))) / t_0);
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(Float64(B_m + C)) * Float64(-sqrt(F))));
	end
	return tmp
end

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 = (B_m ^ 2.0) - ((4.0 * A) * C);
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 5e+92)
		tmp = -sqrt(((2.0 * (t_0 * F)) * ((2.0 * C) + (-0.5 * (B_m * (B_m * (1.0 / A))))))) / t_0;
	else
		tmp = (sqrt(2.0) / B_m) * (sqrt((B_m + C)) * -sqrt(F));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
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[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+92], N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$0 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * C), $MachinePrecision] + N[(-0.5 * N[(B$95$m * N[(B$95$m * N[(1.0 / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]

Derivation

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

Alternative 11: 51.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B_m}^{2} \leq 10^{+94}:\\ \;\;\;\;\frac{-\sqrt{\left(F \cdot t_0\right) \cdot \left(2 \cdot \left(C + C\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B_m} \cdot \left(\sqrt{B_m + C} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= (pow B_m 2.0) 1e+94)
     (/ (- (sqrt (* (* F t_0) (* 2.0 (+ C C))))) t_0)
     (* (/ (sqrt 2.0) B_m) (* (sqrt (+ B_m C)) (- (sqrt F)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (pow(B_m, 2.0) <= 1e+94) {
		tmp = -sqrt(((F * t_0) * (2.0 * (C + C)))) / t_0;
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt((B_m + C)) * -sqrt(F));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e+94)
		tmp = Float64(Float64(-sqrt(Float64(Float64(F * t_0) * Float64(2.0 * Float64(C + C))))) / t_0);
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(Float64(B_m + C)) * Float64(-sqrt(F))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+94], N[((-N[Sqrt[N[(N[(F * t$95$0), $MachinePrecision] * N[(2.0 * N[(C + C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]

Derivation

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

Alternative 12: 47.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B_m \leq 8.5 \cdot 10^{-59}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot F\right)\right)} \cdot \left(-\sqrt{C + C}\right)}{\mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B_m} \cdot \left(\sqrt{B_m + C} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 8.5e-59)
   (/
    (* (sqrt (* -8.0 (* A (* C F)))) (- (sqrt (+ C C))))
    (fma B_m B_m (* A (* C -4.0))))
   (* (/ (sqrt 2.0) B_m) (* (sqrt (+ B_m C)) (- (sqrt F))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 8.5e-59) {
		tmp = (sqrt((-8.0 * (A * (C * F)))) * -sqrt((C + C))) / fma(B_m, B_m, (A * (C * -4.0)));
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt((B_m + C)) * -sqrt(F));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 8.5e-59)
		tmp = Float64(Float64(sqrt(Float64(-8.0 * Float64(A * Float64(C * F)))) * Float64(-sqrt(Float64(C + C)))) / fma(B_m, B_m, Float64(A * Float64(C * -4.0))));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(Float64(B_m + C)) * Float64(-sqrt(F))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 8.5e-59], N[(N[(N[Sqrt[N[(-8.0 * N[(A * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(C + C), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]

Derivation

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

Alternative 13: 37.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B_m \leq 6.8 \cdot 10^{-133}:\\ \;\;\;\;\frac{-\sqrt{\left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right) \cdot \left(2 \cdot \left(B_m + C\right)\right)}}{\mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B_m} \cdot \left(\sqrt{B_m + C} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 6.8e-133)
   (/
    (- (sqrt (* (* -4.0 (* A (* C F))) (* 2.0 (+ B_m C)))))
    (fma B_m B_m (* A (* C -4.0))))
   (* (/ (sqrt 2.0) B_m) (* (sqrt (+ B_m C)) (- (sqrt F))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 6.8e-133) {
		tmp = -sqrt(((-4.0 * (A * (C * F))) * (2.0 * (B_m + C)))) / fma(B_m, B_m, (A * (C * -4.0)));
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt((B_m + C)) * -sqrt(F));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 6.8e-133)
		tmp = Float64(Float64(-sqrt(Float64(Float64(-4.0 * Float64(A * Float64(C * F))) * Float64(2.0 * Float64(B_m + C))))) / fma(B_m, B_m, Float64(A * Float64(C * -4.0))));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(Float64(B_m + C)) * Float64(-sqrt(F))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 6.8e-133], N[((-N[Sqrt[N[(N[(-4.0 * N[(A * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(B$95$m + C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]

Derivation

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

Alternative 14: 35.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;C \leq 8.6 \cdot 10^{+157}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{F} \cdot \frac{-1}{\sqrt{B_m}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{C \cdot F} \cdot \frac{-{\left(\sqrt{2}\right)}^{2}}{B_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= C 8.6e+157)
   (* (sqrt 2.0) (* (sqrt F) (/ -1.0 (sqrt B_m))))
   (* (sqrt (* C F)) (/ (- (pow (sqrt 2.0) 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 <= 8.6e+157) {
		tmp = sqrt(2.0) * (sqrt(F) * (-1.0 / sqrt(B_m)));
	} else {
		tmp = sqrt((C * F)) * (-pow(sqrt(2.0), 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 <= 8.6d+157) then
        tmp = sqrt(2.0d0) * (sqrt(f) * ((-1.0d0) / sqrt(b_m)))
    else
        tmp = sqrt((c * f)) * (-(sqrt(2.0d0) ** 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 <= 8.6e+157) {
		tmp = Math.sqrt(2.0) * (Math.sqrt(F) * (-1.0 / Math.sqrt(B_m)));
	} else {
		tmp = Math.sqrt((C * F)) * (-Math.pow(Math.sqrt(2.0), 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 <= 8.6e+157:
		tmp = math.sqrt(2.0) * (math.sqrt(F) * (-1.0 / math.sqrt(B_m)))
	else:
		tmp = math.sqrt((C * F)) * (-math.pow(math.sqrt(2.0), 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 <= 8.6e+157)
		tmp = Float64(sqrt(2.0) * Float64(sqrt(F) * Float64(-1.0 / sqrt(B_m))));
	else
		tmp = Float64(sqrt(Float64(C * F)) * Float64(Float64(-(sqrt(2.0) ^ 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 <= 8.6e+157)
		tmp = sqrt(2.0) * (sqrt(F) * (-1.0 / sqrt(B_m)));
	else
		tmp = sqrt((C * F)) * (-(sqrt(2.0) ^ 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, 8.6e+157], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * N[(-1.0 / N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision] * N[((-N[Power[N[Sqrt[2.0], $MachinePrecision], 2.0], $MachinePrecision]) / B$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

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

Alternative 15: 35.3% accurate, 2.1× speedup

Math

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

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

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return sqrt(2.0) * (sqrt(F) * (-1.0 / sqrt(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) * (sqrt(f) * ((-1.0d0) / sqrt(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.sqrt(F) * (-1.0 / Math.sqrt(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.sqrt(F) * (-1.0 / math.sqrt(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(2.0) * Float64(sqrt(F) * Float64(-1.0 / sqrt(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) * (sqrt(F) * (-1.0 / sqrt(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[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * N[(-1.0 / N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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

Alternative 16: 35.3% accurate, 2.1× speedup

Math

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

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 F) (sqrt B_m)) (- (sqrt 2.0))))

C

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

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

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

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

Derivation

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

Alternative 17: 34.6% accurate, 5.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;F \leq 780:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(F \cdot \left(B_m + C\right)\right)}}{B_m}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{2 \cdot F}{B_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F 780.0)
   (- (/ (sqrt (* 2.0 (* F (+ B_m C)))) B_m))
   (- (sqrt (/ (* 2.0 F) B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 780.0) {
		tmp = -(sqrt((2.0 * (F * (B_m + C)))) / B_m);
	} else {
		tmp = -sqrt(((2.0 * F) / B_m));
	}
	return tmp;
}

Fortran

B_m = abs(B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (f <= 780.0d0) then
        tmp = -(sqrt((2.0d0 * (f * (b_m + c)))) / b_m)
    else
        tmp = -sqrt(((2.0d0 * f) / b_m))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 780.0) {
		tmp = -(Math.sqrt((2.0 * (F * (B_m + C)))) / B_m);
	} else {
		tmp = -Math.sqrt(((2.0 * F) / B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if F <= 780.0:
		tmp = -(math.sqrt((2.0 * (F * (B_m + C)))) / B_m)
	else:
		tmp = -math.sqrt(((2.0 * F) / B_m))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= 780.0)
		tmp = Float64(-Float64(sqrt(Float64(2.0 * Float64(F * Float64(B_m + C)))) / B_m));
	else
		tmp = Float64(-sqrt(Float64(Float64(2.0 * F) / B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= 780.0)
		tmp = -(sqrt((2.0 * (F * (B_m + C)))) / B_m);
	else
		tmp = -sqrt(((2.0 * F) / B_m));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, 780.0], (-N[(N[Sqrt[N[(2.0 * N[(F * N[(B$95$m + C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision]), (-N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] / B$95$m), $MachinePrecision]], $MachinePrecision])]

Derivation

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

Alternative 18: 27.1% accurate, 6.0× speedup

Math

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

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

Derivation

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

Alternative 19: 5.2% accurate, 634.0× speedup

Math

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

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 -1.0)

C

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

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 = -1.0d0
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 -1.0;
}

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 -1.0

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return -1.0
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 = -1.0;
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_] := -1.0

Derivation

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

Reproduce

herbie shell --seed 2024006 
(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))))