ABCF->ab-angle a

Percentage Accurate: 19.5% → 56.7%

Time: 37.6s

Alternatives: 14

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: 19.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 56.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := 2 \cdot \left(C + C\right)\\ t_1 := \sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B_m, A - C\right)\right)\right)}\\ t_2 := \mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)\\ t_3 := F \cdot t_2\\ t_4 := -\frac{\sqrt{\left(2 \cdot \left(C + \left(C + -0.5 \cdot \frac{{B_m}^{2}}{A}\right)\right)\right) \cdot t_3}}{t_2}\\ t_5 := \mathsf{fma}\left(A, C \cdot -4, {B_m}^{2}\right)\\ \mathbf{if}\;B_m \leq 4.8 \cdot 10^{-98}:\\ \;\;\;\;\frac{-\sqrt{t_3 \cdot t_0}}{t_2}\\ \mathbf{elif}\;B_m \leq 3.1 \cdot 10^{-63}:\\ \;\;\;\;\frac{\sqrt{F \cdot t_5} \cdot \left(-\sqrt{t_0}\right)}{t_2}\\ \mathbf{elif}\;B_m \leq 1.02 \cdot 10^{-5}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B_m \leq 2.95 \cdot 10^{+41}:\\ \;\;\;\;\frac{\sqrt{F}}{\sqrt{t_5}} \cdot \left(-t_1\right)\\ \mathbf{elif}\;B_m \leq 2.12 \cdot 10^{+46}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B_m \leq 7.6 \cdot 10^{+233}:\\ \;\;\;\;t_1 \cdot \left(\sqrt{F} \cdot \frac{-1}{B_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\frac{\sqrt{B_m}}{\sqrt{2}}}\\ \end{array} \end{array} \]

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 (* 2.0 (+ C C)))
        (t_1 (sqrt (* 2.0 (+ C (+ A (hypot B_m (- A C)))))))
        (t_2 (fma B_m B_m (* A (* C -4.0))))
        (t_3 (* F t_2))
        (t_4
         (-
          (/
           (sqrt (* (* 2.0 (+ C (+ C (* -0.5 (/ (pow B_m 2.0) A))))) t_3))
           t_2)))
        (t_5 (fma A (* C -4.0) (pow B_m 2.0))))
   (if (<= B_m 4.8e-98)
     (/ (- (sqrt (* t_3 t_0))) t_2)
     (if (<= B_m 3.1e-63)
       (/ (* (sqrt (* F t_5)) (- (sqrt t_0))) t_2)
       (if (<= B_m 1.02e-5)
         t_4
         (if (<= B_m 2.95e+41)
           (* (/ (sqrt F) (sqrt t_5)) (- t_1))
           (if (<= B_m 2.12e+46)
             t_4
             (if (<= B_m 7.6e+233)
               (* t_1 (* (sqrt F) (/ -1.0 B_m)))
               (/ (- (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) {
	double t_0 = 2.0 * (C + C);
	double t_1 = sqrt((2.0 * (C + (A + hypot(B_m, (A - C))))));
	double t_2 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_3 = F * t_2;
	double t_4 = -(sqrt(((2.0 * (C + (C + (-0.5 * (pow(B_m, 2.0) / A))))) * t_3)) / t_2);
	double t_5 = fma(A, (C * -4.0), pow(B_m, 2.0));
	double tmp;
	if (B_m <= 4.8e-98) {
		tmp = -sqrt((t_3 * t_0)) / t_2;
	} else if (B_m <= 3.1e-63) {
		tmp = (sqrt((F * t_5)) * -sqrt(t_0)) / t_2;
	} else if (B_m <= 1.02e-5) {
		tmp = t_4;
	} else if (B_m <= 2.95e+41) {
		tmp = (sqrt(F) / sqrt(t_5)) * -t_1;
	} else if (B_m <= 2.12e+46) {
		tmp = t_4;
	} else if (B_m <= 7.6e+233) {
		tmp = t_1 * (sqrt(F) * (-1.0 / B_m));
	} else {
		tmp = -sqrt(F) / (sqrt(B_m) / sqrt(2.0));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(2.0 * Float64(C + C))
	t_1 = sqrt(Float64(2.0 * 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(F * t_2)
	t_4 = Float64(-Float64(sqrt(Float64(Float64(2.0 * Float64(C + Float64(C + Float64(-0.5 * Float64((B_m ^ 2.0) / A))))) * t_3)) / t_2))
	t_5 = fma(A, Float64(C * -4.0), (B_m ^ 2.0))
	tmp = 0.0
	if (B_m <= 4.8e-98)
		tmp = Float64(Float64(-sqrt(Float64(t_3 * t_0))) / t_2);
	elseif (B_m <= 3.1e-63)
		tmp = Float64(Float64(sqrt(Float64(F * t_5)) * Float64(-sqrt(t_0))) / t_2);
	elseif (B_m <= 1.02e-5)
		tmp = t_4;
	elseif (B_m <= 2.95e+41)
		tmp = Float64(Float64(sqrt(F) / sqrt(t_5)) * Float64(-t_1));
	elseif (B_m <= 2.12e+46)
		tmp = t_4;
	elseif (B_m <= 7.6e+233)
		tmp = Float64(t_1 * Float64(sqrt(F) * Float64(-1.0 / B_m)));
	else
		tmp = Float64(Float64(-sqrt(F)) / Float64(sqrt(B_m) / sqrt(2.0)));
	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[(2.0 * N[(C + C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(2.0 * N[(C + N[(A + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $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[(F * t$95$2), $MachinePrecision]}, Block[{t$95$4 = (-N[(N[Sqrt[N[(N[(2.0 * N[(C + N[(C + N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision])}, Block[{t$95$5 = N[(A * N[(C * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 4.8e-98], N[((-N[Sqrt[N[(t$95$3 * t$95$0), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision], If[LessEqual[B$95$m, 3.1e-63], N[(N[(N[Sqrt[N[(F * t$95$5), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[t$95$0], $MachinePrecision])), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[B$95$m, 1.02e-5], t$95$4, If[LessEqual[B$95$m, 2.95e+41], N[(N[(N[Sqrt[F], $MachinePrecision] / N[Sqrt[t$95$5], $MachinePrecision]), $MachinePrecision] * (-t$95$1)), $MachinePrecision], If[LessEqual[B$95$m, 2.12e+46], t$95$4, If[LessEqual[B$95$m, 7.6e+233], N[(t$95$1 * N[(N[Sqrt[F], $MachinePrecision] * N[(-1.0 / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) / N[(N[Sqrt[B$95$m], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]

Derivation

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

Alternative 2: 60.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, {B_m}^{2}\right)\\ t_1 := {B_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{-\sqrt{\left(2 \cdot \left(t_1 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t_1}\\ \mathbf{if}\;t_2 \leq -5 \cdot 10^{-173}:\\ \;\;\;\;\frac{\sqrt{F}}{\sqrt{t_0}} \cdot \left(-\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B_m, A - C\right)\right)\right)}\right)\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;\frac{\sqrt{F \cdot t_0} \cdot \left(-\sqrt{2 \cdot \left(C + \left(C + -0.5 \cdot \frac{{B_m}^{2}}{A}\right)\right)}\right)}{\mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\frac{\sqrt{B_m}}{\sqrt{2}}}\\ \end{array} \end{array} \]

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 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_2
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_1 F))
             (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0)))))))
          t_1)))
   (if (<= t_2 -5e-173)
     (*
      (/ (sqrt F) (sqrt t_0))
      (- (sqrt (* 2.0 (+ C (+ A (hypot B_m (- A C))))))))
     (if (<= t_2 INFINITY)
       (/
        (*
         (sqrt (* F t_0))
         (- (sqrt (* 2.0 (+ C (+ C (* -0.5 (/ (pow B_m 2.0) A))))))))
        (fma B_m B_m (* A (* C -4.0))))
       (/ (- (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) {
	double t_0 = fma(A, (C * -4.0), pow(B_m, 2.0));
	double t_1 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_2 = -sqrt(((2.0 * (t_1 * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / t_1;
	double tmp;
	if (t_2 <= -5e-173) {
		tmp = (sqrt(F) / sqrt(t_0)) * -sqrt((2.0 * (C + (A + hypot(B_m, (A - C))))));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = (sqrt((F * t_0)) * -sqrt((2.0 * (C + (C + (-0.5 * (pow(B_m, 2.0) / A))))))) / fma(B_m, B_m, (A * (C * -4.0)));
	} else {
		tmp = -sqrt(F) / (sqrt(B_m) / sqrt(2.0));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(A, Float64(C * -4.0), (B_m ^ 2.0))
	t_1 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_2 = 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)
	tmp = 0.0
	if (t_2 <= -5e-173)
		tmp = Float64(Float64(sqrt(F) / sqrt(t_0)) * Float64(-sqrt(Float64(2.0 * Float64(C + Float64(A + hypot(B_m, Float64(A - C))))))));
	elseif (t_2 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(F * t_0)) * Float64(-sqrt(Float64(2.0 * Float64(C + Float64(C + Float64(-0.5 * Float64((B_m ^ 2.0) / A)))))))) / fma(B_m, B_m, Float64(A * Float64(C * -4.0))));
	else
		tmp = Float64(Float64(-sqrt(F)) / Float64(sqrt(B_m) / sqrt(2.0)));
	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[(A * N[(C * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $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[((-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]}, If[LessEqual[t$95$2, -5e-173], N[(N[(N[Sqrt[F], $MachinePrecision] / N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] * (-N[Sqrt[N[(2.0 * N[(C + N[(A + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[(N[Sqrt[N[(F * t$95$0), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * N[(C + N[(C + N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) / N[(N[Sqrt[B$95$m], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

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

Alternative 3: 55.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B_m, A - C\right)\right)\right)} \cdot \left(\sqrt{F} \cdot \frac{-1}{B_m}\right)\\ t_1 := 2 \cdot \left(C + C\right)\\ t_2 := \mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)\\ t_3 := F \cdot t_2\\ t_4 := -\frac{\sqrt{\left(2 \cdot \left(C + \left(C + -0.5 \cdot \frac{{B_m}^{2}}{A}\right)\right)\right) \cdot t_3}}{t_2}\\ \mathbf{if}\;B_m \leq 4.5 \cdot 10^{-97}:\\ \;\;\;\;\frac{-\sqrt{t_3 \cdot t_1}}{t_2}\\ \mathbf{elif}\;B_m \leq 5.2 \cdot 10^{-63}:\\ \;\;\;\;\frac{\sqrt{F \cdot \mathsf{fma}\left(A, C \cdot -4, {B_m}^{2}\right)} \cdot \left(-\sqrt{t_1}\right)}{t_2}\\ \mathbf{elif}\;B_m \leq 3.1 \cdot 10^{+21}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B_m \leq 2.35 \cdot 10^{+41}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B_m \leq 2.12 \cdot 10^{+46}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B_m \leq 9.5 \cdot 10^{+233}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\frac{\sqrt{B_m}}{\sqrt{2}}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0
         (*
          (sqrt (* 2.0 (+ C (+ A (hypot B_m (- A C))))))
          (* (sqrt F) (/ -1.0 B_m))))
        (t_1 (* 2.0 (+ C C)))
        (t_2 (fma B_m B_m (* A (* C -4.0))))
        (t_3 (* F t_2))
        (t_4
         (-
          (/
           (sqrt (* (* 2.0 (+ C (+ C (* -0.5 (/ (pow B_m 2.0) A))))) t_3))
           t_2))))
   (if (<= B_m 4.5e-97)
     (/ (- (sqrt (* t_3 t_1))) t_2)
     (if (<= B_m 5.2e-63)
       (/ (* (sqrt (* F (fma A (* C -4.0) (pow B_m 2.0)))) (- (sqrt t_1))) t_2)
       (if (<= B_m 3.1e+21)
         t_4
         (if (<= B_m 2.35e+41)
           t_0
           (if (<= B_m 2.12e+46)
             t_4
             (if (<= B_m 9.5e+233)
               t_0
               (/ (- (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) {
	double t_0 = sqrt((2.0 * (C + (A + hypot(B_m, (A - C)))))) * (sqrt(F) * (-1.0 / B_m));
	double t_1 = 2.0 * (C + C);
	double t_2 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_3 = F * t_2;
	double t_4 = -(sqrt(((2.0 * (C + (C + (-0.5 * (pow(B_m, 2.0) / A))))) * t_3)) / t_2);
	double tmp;
	if (B_m <= 4.5e-97) {
		tmp = -sqrt((t_3 * t_1)) / t_2;
	} else if (B_m <= 5.2e-63) {
		tmp = (sqrt((F * fma(A, (C * -4.0), pow(B_m, 2.0)))) * -sqrt(t_1)) / t_2;
	} else if (B_m <= 3.1e+21) {
		tmp = t_4;
	} else if (B_m <= 2.35e+41) {
		tmp = t_0;
	} else if (B_m <= 2.12e+46) {
		tmp = t_4;
	} else if (B_m <= 9.5e+233) {
		tmp = t_0;
	} else {
		tmp = -sqrt(F) / (sqrt(B_m) / sqrt(2.0));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(sqrt(Float64(2.0 * Float64(C + Float64(A + hypot(B_m, Float64(A - C)))))) * Float64(sqrt(F) * Float64(-1.0 / B_m)))
	t_1 = Float64(2.0 * Float64(C + C))
	t_2 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_3 = Float64(F * t_2)
	t_4 = Float64(-Float64(sqrt(Float64(Float64(2.0 * Float64(C + Float64(C + Float64(-0.5 * Float64((B_m ^ 2.0) / A))))) * t_3)) / t_2))
	tmp = 0.0
	if (B_m <= 4.5e-97)
		tmp = Float64(Float64(-sqrt(Float64(t_3 * t_1))) / t_2);
	elseif (B_m <= 5.2e-63)
		tmp = Float64(Float64(sqrt(Float64(F * fma(A, Float64(C * -4.0), (B_m ^ 2.0)))) * Float64(-sqrt(t_1))) / t_2);
	elseif (B_m <= 3.1e+21)
		tmp = t_4;
	elseif (B_m <= 2.35e+41)
		tmp = t_0;
	elseif (B_m <= 2.12e+46)
		tmp = t_4;
	elseif (B_m <= 9.5e+233)
		tmp = t_0;
	else
		tmp = Float64(Float64(-sqrt(F)) / Float64(sqrt(B_m) / sqrt(2.0)));
	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[(N[Sqrt[N[(2.0 * N[(C + N[(A + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * N[(-1.0 / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[(C + C), $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[(F * t$95$2), $MachinePrecision]}, Block[{t$95$4 = (-N[(N[Sqrt[N[(N[(2.0 * N[(C + N[(C + N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision])}, If[LessEqual[B$95$m, 4.5e-97], N[((-N[Sqrt[N[(t$95$3 * t$95$1), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision], If[LessEqual[B$95$m, 5.2e-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[t$95$1], $MachinePrecision])), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[B$95$m, 3.1e+21], t$95$4, If[LessEqual[B$95$m, 2.35e+41], t$95$0, If[LessEqual[B$95$m, 2.12e+46], t$95$4, If[LessEqual[B$95$m, 9.5e+233], t$95$0, N[((-N[Sqrt[F], $MachinePrecision]) / N[(N[Sqrt[B$95$m], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

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

Alternative 4: 55.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B_m, A - C\right)\right)\right)} \cdot \left(\sqrt{F} \cdot \frac{-1}{B_m}\right)\\ t_1 := \mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)\\ t_2 := -\frac{\sqrt{\left(2 \cdot \left(C + \left(C + -0.5 \cdot \frac{{B_m}^{2}}{A}\right)\right)\right) \cdot \left(F \cdot t_1\right)}}{t_1}\\ \mathbf{if}\;B_m \leq 3.9 \cdot 10^{+21}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B_m \leq 2.5 \cdot 10^{+41}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B_m \leq 2.3 \cdot 10^{+46}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B_m \leq 7.5 \cdot 10^{+233}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\frac{\sqrt{B_m}}{\sqrt{2}}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0
         (*
          (sqrt (* 2.0 (+ C (+ A (hypot B_m (- A C))))))
          (* (sqrt F) (/ -1.0 B_m))))
        (t_1 (fma B_m B_m (* A (* C -4.0))))
        (t_2
         (-
          (/
           (sqrt
            (* (* 2.0 (+ C (+ C (* -0.5 (/ (pow B_m 2.0) A))))) (* F t_1)))
           t_1))))
   (if (<= B_m 3.9e+21)
     t_2
     (if (<= B_m 2.5e+41)
       t_0
       (if (<= B_m 2.3e+46)
         t_2
         (if (<= B_m 7.5e+233)
           t_0
           (/ (- (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) {
	double t_0 = sqrt((2.0 * (C + (A + hypot(B_m, (A - C)))))) * (sqrt(F) * (-1.0 / B_m));
	double t_1 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_2 = -(sqrt(((2.0 * (C + (C + (-0.5 * (pow(B_m, 2.0) / A))))) * (F * t_1))) / t_1);
	double tmp;
	if (B_m <= 3.9e+21) {
		tmp = t_2;
	} else if (B_m <= 2.5e+41) {
		tmp = t_0;
	} else if (B_m <= 2.3e+46) {
		tmp = t_2;
	} else if (B_m <= 7.5e+233) {
		tmp = t_0;
	} else {
		tmp = -sqrt(F) / (sqrt(B_m) / sqrt(2.0));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(sqrt(Float64(2.0 * Float64(C + Float64(A + hypot(B_m, Float64(A - C)))))) * Float64(sqrt(F) * Float64(-1.0 / B_m)))
	t_1 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_2 = Float64(-Float64(sqrt(Float64(Float64(2.0 * Float64(C + Float64(C + Float64(-0.5 * Float64((B_m ^ 2.0) / A))))) * Float64(F * t_1))) / t_1))
	tmp = 0.0
	if (B_m <= 3.9e+21)
		tmp = t_2;
	elseif (B_m <= 2.5e+41)
		tmp = t_0;
	elseif (B_m <= 2.3e+46)
		tmp = t_2;
	elseif (B_m <= 7.5e+233)
		tmp = t_0;
	else
		tmp = Float64(Float64(-sqrt(F)) / Float64(sqrt(B_m) / sqrt(2.0)));
	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[(N[Sqrt[N[(2.0 * N[(C + N[(A + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * N[(-1.0 / B$95$m), $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[N[(N[(2.0 * N[(C + N[(C + N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(F * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision])}, If[LessEqual[B$95$m, 3.9e+21], t$95$2, If[LessEqual[B$95$m, 2.5e+41], t$95$0, If[LessEqual[B$95$m, 2.3e+46], t$95$2, If[LessEqual[B$95$m, 7.5e+233], t$95$0, N[((-N[Sqrt[F], $MachinePrecision]) / N[(N[Sqrt[B$95$m], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

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

Alternative 5: 56.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := {B_m}^{2} - \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;B_m \leq 2.8 \cdot 10^{+20}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(t_0 \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{t_0}\\ \mathbf{elif}\;B_m \leq 8.2 \cdot 10^{+233}:\\ \;\;\;\;\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B_m, A - C\right)\right)\right)} \cdot \left(\sqrt{F} \cdot \frac{-1}{B_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\frac{\sqrt{B_m}}{\sqrt{2}}}\\ \end{array} \end{array} \]

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 (<= B_m 2.8e+20)
     (/ (- (sqrt (* (* 2.0 (* t_0 F)) (* 2.0 C)))) t_0)
     (if (<= B_m 8.2e+233)
       (*
        (sqrt (* 2.0 (+ C (+ A (hypot B_m (- A C))))))
        (* (sqrt F) (/ -1.0 B_m)))
       (/ (- (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) {
	double t_0 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double tmp;
	if (B_m <= 2.8e+20) {
		tmp = -sqrt(((2.0 * (t_0 * F)) * (2.0 * C))) / t_0;
	} else if (B_m <= 8.2e+233) {
		tmp = sqrt((2.0 * (C + (A + hypot(B_m, (A - C)))))) * (sqrt(F) * (-1.0 / B_m));
	} else {
		tmp = -sqrt(F) / (sqrt(B_m) / sqrt(2.0));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = Math.pow(B_m, 2.0) - ((4.0 * A) * C);
	double tmp;
	if (B_m <= 2.8e+20) {
		tmp = -Math.sqrt(((2.0 * (t_0 * F)) * (2.0 * C))) / t_0;
	} else if (B_m <= 8.2e+233) {
		tmp = Math.sqrt((2.0 * (C + (A + Math.hypot(B_m, (A - C)))))) * (Math.sqrt(F) * (-1.0 / B_m));
	} else {
		tmp = -Math.sqrt(F) / (Math.sqrt(B_m) / Math.sqrt(2.0));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = math.pow(B_m, 2.0) - ((4.0 * A) * C)
	tmp = 0
	if B_m <= 2.8e+20:
		tmp = -math.sqrt(((2.0 * (t_0 * F)) * (2.0 * C))) / t_0
	elif B_m <= 8.2e+233:
		tmp = math.sqrt((2.0 * (C + (A + math.hypot(B_m, (A - C)))))) * (math.sqrt(F) * (-1.0 / B_m))
	else:
		tmp = -math.sqrt(F) / (math.sqrt(B_m) / math.sqrt(2.0))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	tmp = 0.0
	if (B_m <= 2.8e+20)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_0 * F)) * Float64(2.0 * C)))) / t_0);
	elseif (B_m <= 8.2e+233)
		tmp = Float64(sqrt(Float64(2.0 * Float64(C + Float64(A + hypot(B_m, Float64(A - C)))))) * Float64(sqrt(F) * Float64(-1.0 / B_m)));
	else
		tmp = Float64(Float64(-sqrt(F)) / Float64(sqrt(B_m) / sqrt(2.0)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = (B_m ^ 2.0) - ((4.0 * A) * C);
	tmp = 0.0;
	if (B_m <= 2.8e+20)
		tmp = -sqrt(((2.0 * (t_0 * F)) * (2.0 * C))) / t_0;
	elseif (B_m <= 8.2e+233)
		tmp = sqrt((2.0 * (C + (A + hypot(B_m, (A - C)))))) * (sqrt(F) * (-1.0 / B_m));
	else
		tmp = -sqrt(F) / (sqrt(B_m) / sqrt(2.0));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 2.8e+20], N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$0 * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[B$95$m, 8.2e+233], N[(N[Sqrt[N[(2.0 * N[(C + N[(A + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * N[(-1.0 / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) / N[(N[Sqrt[B$95$m], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

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

Alternative 6: 41.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B_m \leq 8.5 \cdot 10^{-71}:\\ \;\;\;\;-\frac{\sqrt{\left(F \cdot t_0\right) \cdot \left(2 \cdot \left(B_m + C\right)\right)}}{t_0}\\ \mathbf{elif}\;B_m \leq 7.6 \cdot 10^{+233}:\\ \;\;\;\;\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B_m, A - C\right)\right)\right)} \cdot \left(\sqrt{F} \cdot \frac{-1}{B_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\frac{\sqrt{B_m}}{\sqrt{2}}}\\ \end{array} \end{array} \]

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 (<= B_m 8.5e-71)
     (- (/ (sqrt (* (* F t_0) (* 2.0 (+ B_m C)))) t_0))
     (if (<= B_m 7.6e+233)
       (*
        (sqrt (* 2.0 (+ C (+ A (hypot B_m (- A C))))))
        (* (sqrt F) (/ -1.0 B_m)))
       (/ (- (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) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (B_m <= 8.5e-71) {
		tmp = -(sqrt(((F * t_0) * (2.0 * (B_m + C)))) / t_0);
	} else if (B_m <= 7.6e+233) {
		tmp = sqrt((2.0 * (C + (A + hypot(B_m, (A - C)))))) * (sqrt(F) * (-1.0 / B_m));
	} else {
		tmp = -sqrt(F) / (sqrt(B_m) / sqrt(2.0));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if (B_m <= 8.5e-71)
		tmp = Float64(-Float64(sqrt(Float64(Float64(F * t_0) * Float64(2.0 * Float64(B_m + C)))) / t_0));
	elseif (B_m <= 7.6e+233)
		tmp = Float64(sqrt(Float64(2.0 * Float64(C + Float64(A + hypot(B_m, Float64(A - C)))))) * Float64(sqrt(F) * Float64(-1.0 / B_m)));
	else
		tmp = Float64(Float64(-sqrt(F)) / Float64(sqrt(B_m) / sqrt(2.0)));
	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[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 8.5e-71], (-N[(N[Sqrt[N[(N[(F * t$95$0), $MachinePrecision] * N[(2.0 * N[(B$95$m + C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), If[LessEqual[B$95$m, 7.6e+233], N[(N[Sqrt[N[(2.0 * N[(C + N[(A + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * N[(-1.0 / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) / N[(N[Sqrt[B$95$m], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

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

Alternative 7: 56.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B_m \leq 3.5 \cdot 10^{+19}:\\ \;\;\;\;\frac{-\sqrt{\left(F \cdot t_0\right) \cdot \left(2 \cdot \left(C + C\right)\right)}}{t_0}\\ \mathbf{elif}\;B_m \leq 1.26 \cdot 10^{+234}:\\ \;\;\;\;\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B_m, A - C\right)\right)\right)} \cdot \left(\sqrt{F} \cdot \frac{-1}{B_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\frac{\sqrt{B_m}}{\sqrt{2}}}\\ \end{array} \end{array} \]

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 (<= B_m 3.5e+19)
     (/ (- (sqrt (* (* F t_0) (* 2.0 (+ C C))))) t_0)
     (if (<= B_m 1.26e+234)
       (*
        (sqrt (* 2.0 (+ C (+ A (hypot B_m (- A C))))))
        (* (sqrt F) (/ -1.0 B_m)))
       (/ (- (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) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (B_m <= 3.5e+19) {
		tmp = -sqrt(((F * t_0) * (2.0 * (C + C)))) / t_0;
	} else if (B_m <= 1.26e+234) {
		tmp = sqrt((2.0 * (C + (A + hypot(B_m, (A - C)))))) * (sqrt(F) * (-1.0 / B_m));
	} else {
		tmp = -sqrt(F) / (sqrt(B_m) / sqrt(2.0));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if (B_m <= 3.5e+19)
		tmp = Float64(Float64(-sqrt(Float64(Float64(F * t_0) * Float64(2.0 * Float64(C + C))))) / t_0);
	elseif (B_m <= 1.26e+234)
		tmp = Float64(sqrt(Float64(2.0 * Float64(C + Float64(A + hypot(B_m, Float64(A - C)))))) * Float64(sqrt(F) * Float64(-1.0 / B_m)));
	else
		tmp = Float64(Float64(-sqrt(F)) / Float64(sqrt(B_m) / sqrt(2.0)));
	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[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 3.5e+19], N[((-N[Sqrt[N[(N[(F * t$95$0), $MachinePrecision] * N[(2.0 * N[(C + C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[B$95$m, 1.26e+234], N[(N[Sqrt[N[(2.0 * N[(C + N[(A + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * N[(-1.0 / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) / N[(N[Sqrt[B$95$m], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

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

Alternative 8: 41.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B_m \leq 5 \cdot 10^{-120}:\\ \;\;\;\;-\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(B_m + C\right)\right)}}{C \cdot \left(A \cdot -4\right)}\\ \mathbf{elif}\;B_m \leq 1.2 \cdot 10^{+234}:\\ \;\;\;\;\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B_m, A - C\right)\right)\right)} \cdot \left(\sqrt{F} \cdot \frac{-1}{B_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\frac{\sqrt{B_m}}{\sqrt{2}}}\\ \end{array} \end{array} \]

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 5e-120)
   (-
    (/
     (sqrt (* (* F (fma B_m B_m (* A (* C -4.0)))) (* 2.0 (+ B_m C))))
     (* C (* A -4.0))))
   (if (<= B_m 1.2e+234)
     (*
      (sqrt (* 2.0 (+ C (+ A (hypot B_m (- A C))))))
      (* (sqrt F) (/ -1.0 B_m)))
     (/ (- (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) {
	double tmp;
	if (B_m <= 5e-120) {
		tmp = -(sqrt(((F * fma(B_m, B_m, (A * (C * -4.0)))) * (2.0 * (B_m + C)))) / (C * (A * -4.0)));
	} else if (B_m <= 1.2e+234) {
		tmp = sqrt((2.0 * (C + (A + hypot(B_m, (A - C)))))) * (sqrt(F) * (-1.0 / B_m));
	} else {
		tmp = -sqrt(F) / (sqrt(B_m) / sqrt(2.0));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 5e-120)
		tmp = Float64(-Float64(sqrt(Float64(Float64(F * fma(B_m, B_m, Float64(A * Float64(C * -4.0)))) * Float64(2.0 * Float64(B_m + C)))) / Float64(C * Float64(A * -4.0))));
	elseif (B_m <= 1.2e+234)
		tmp = Float64(sqrt(Float64(2.0 * Float64(C + Float64(A + hypot(B_m, Float64(A - C)))))) * Float64(sqrt(F) * Float64(-1.0 / B_m)));
	else
		tmp = Float64(Float64(-sqrt(F)) / Float64(sqrt(B_m) / sqrt(2.0)));
	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_] := If[LessEqual[B$95$m, 5e-120], (-N[(N[Sqrt[N[(N[(F * N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(B$95$m + C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), If[LessEqual[B$95$m, 1.2e+234], N[(N[Sqrt[N[(2.0 * N[(C + N[(A + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * N[(-1.0 / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) / N[(N[Sqrt[B$95$m], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

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

Alternative 9: 36.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B_m \leq 4.1 \cdot 10^{-120}:\\ \;\;\;\;-\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(B_m + C\right)\right)}}{C \cdot \left(A \cdot -4\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{-\sqrt{F}}{\sqrt{B_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 4.1e-120)
   (-
    (/
     (sqrt (* (* F (fma B_m B_m (* A (* C -4.0)))) (* 2.0 (+ B_m C))))
     (* C (* A -4.0))))
   (* (sqrt 2.0) (/ (- (sqrt F)) (sqrt B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 4.1e-120) {
		tmp = -(sqrt(((F * fma(B_m, B_m, (A * (C * -4.0)))) * (2.0 * (B_m + C)))) / (C * (A * -4.0)));
	} else {
		tmp = sqrt(2.0) * (-sqrt(F) / sqrt(B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 4.1e-120)
		tmp = Float64(-Float64(sqrt(Float64(Float64(F * fma(B_m, B_m, Float64(A * Float64(C * -4.0)))) * Float64(2.0 * Float64(B_m + C)))) / Float64(C * Float64(A * -4.0))));
	else
		tmp = Float64(sqrt(2.0) * Float64(Float64(-sqrt(F)) / sqrt(B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 4.1e-120], (-N[(N[Sqrt[N[(N[(F * N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(B$95$m + C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), N[(N[Sqrt[2.0], $MachinePrecision] * N[((-N[Sqrt[F], $MachinePrecision]) / N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

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

Alternative 10: 36.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B_m \leq 1.9 \cdot 10^{-117}:\\ \;\;\;\;-\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(B_m + C\right)\right)}}{C \cdot \left(A \cdot -4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\frac{\sqrt{B_m}}{\sqrt{2}}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 1.9e-117)
   (-
    (/
     (sqrt (* (* F (fma B_m B_m (* A (* C -4.0)))) (* 2.0 (+ B_m C))))
     (* C (* A -4.0))))
   (/ (- (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) {
	double tmp;
	if (B_m <= 1.9e-117) {
		tmp = -(sqrt(((F * fma(B_m, B_m, (A * (C * -4.0)))) * (2.0 * (B_m + C)))) / (C * (A * -4.0)));
	} else {
		tmp = -sqrt(F) / (sqrt(B_m) / sqrt(2.0));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 1.9e-117)
		tmp = Float64(-Float64(sqrt(Float64(Float64(F * fma(B_m, B_m, Float64(A * Float64(C * -4.0)))) * Float64(2.0 * Float64(B_m + C)))) / Float64(C * Float64(A * -4.0))));
	else
		tmp = Float64(Float64(-sqrt(F)) / Float64(sqrt(B_m) / sqrt(2.0)));
	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_] := If[LessEqual[B$95$m, 1.9e-117], (-N[(N[Sqrt[N[(N[(F * N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(B$95$m + C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), N[((-N[Sqrt[F], $MachinePrecision]) / N[(N[Sqrt[B$95$m], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

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

Alternative 11: 33.2% accurate, 3.0× speedup

Math

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

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 1e-32)
   (* (/ (sqrt 2.0) B_m) (- (sqrt (* F (+ B_m A)))))
   (* (sqrt 2.0) (/ -1.0 (sqrt (/ B_m F))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 1e-32) {
		tmp = (sqrt(2.0) / B_m) * -sqrt((F * (B_m + A)));
	} else {
		tmp = sqrt(2.0) * (-1.0 / sqrt((B_m / F)));
	}
	return tmp;
}

Fortran

B_m = abs(B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (f <= 1d-32) then
        tmp = (sqrt(2.0d0) / b_m) * -sqrt((f * (b_m + a)))
    else
        tmp = sqrt(2.0d0) * ((-1.0d0) / sqrt((b_m / f)))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 1e-32) {
		tmp = (Math.sqrt(2.0) / B_m) * -Math.sqrt((F * (B_m + A)));
	} else {
		tmp = Math.sqrt(2.0) * (-1.0 / Math.sqrt((B_m / F)));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if F <= 1e-32:
		tmp = (math.sqrt(2.0) / B_m) * -math.sqrt((F * (B_m + A)))
	else:
		tmp = math.sqrt(2.0) * (-1.0 / math.sqrt((B_m / 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 (F <= 1e-32)
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(-sqrt(Float64(F * Float64(B_m + A)))));
	else
		tmp = Float64(sqrt(2.0) * Float64(-1.0 / sqrt(Float64(B_m / F))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= 1e-32)
		tmp = (sqrt(2.0) / B_m) * -sqrt((F * (B_m + A)));
	else
		tmp = sqrt(2.0) * (-1.0 / sqrt((B_m / F)));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, 1e-32], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[N[(F * N[(B$95$m + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(-1.0 / N[Sqrt[N[(B$95$m / F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

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

Alternative 12: 34.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;F \leq 0.031:\\ \;\;\;\;-\frac{\sqrt{2}}{B_m} \cdot \sqrt{F \cdot \left(B_m + C\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{-1}{\sqrt{\frac{B_m}{F}}}\\ \end{array} \end{array} \]

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 0.031)
   (- (* (/ (sqrt 2.0) B_m) (sqrt (* F (+ B_m C)))))
   (* (sqrt 2.0) (/ -1.0 (sqrt (/ B_m F))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 0.031) {
		tmp = -((sqrt(2.0) / B_m) * sqrt((F * (B_m + C))));
	} else {
		tmp = sqrt(2.0) * (-1.0 / sqrt((B_m / F)));
	}
	return tmp;
}

Fortran

B_m = abs(B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (f <= 0.031d0) then
        tmp = -((sqrt(2.0d0) / b_m) * sqrt((f * (b_m + c))))
    else
        tmp = sqrt(2.0d0) * ((-1.0d0) / sqrt((b_m / f)))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 0.031) {
		tmp = -((Math.sqrt(2.0) / B_m) * Math.sqrt((F * (B_m + C))));
	} else {
		tmp = Math.sqrt(2.0) * (-1.0 / Math.sqrt((B_m / F)));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if F <= 0.031:
		tmp = -((math.sqrt(2.0) / B_m) * math.sqrt((F * (B_m + C))))
	else:
		tmp = math.sqrt(2.0) * (-1.0 / math.sqrt((B_m / 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 (F <= 0.031)
		tmp = Float64(-Float64(Float64(sqrt(2.0) / B_m) * sqrt(Float64(F * Float64(B_m + C)))));
	else
		tmp = Float64(sqrt(2.0) * Float64(-1.0 / sqrt(Float64(B_m / F))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= 0.031)
		tmp = -((sqrt(2.0) / B_m) * sqrt((F * (B_m + C))));
	else
		tmp = sqrt(2.0) * (-1.0 / sqrt((B_m / F)));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, 0.031], (-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[2.0], $MachinePrecision] * N[(-1.0 / N[Sqrt[N[(B$95$m / F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

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

Alternative 13: 26.9% accurate, 5.9× speedup

Math

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

FPCore

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

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return -pow((2.0 * (F / B_m)), 0.5);
}

Fortran

B_m = abs(B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = -((2.0d0 * (f / b_m)) ** 0.5d0)
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	return -Math.pow((2.0 * (F / B_m)), 0.5);
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return -math.pow((2.0 * (F / B_m)), 0.5)

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(-(Float64(2.0 * Float64(F / B_m)) ^ 0.5))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = -((2.0 * (F / B_m)) ^ 0.5);
end

Wolfram

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

Derivation

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

Alternative 14: 26.9% accurate, 6.0× speedup

Math

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

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

Reproduce

herbie shell --seed 2024010 
(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))))