ABCF->ab-angle b

Percentage Accurate: 19.1% → 51.9%

Time: 30.5s

Alternatives: 12

Speedup: 3.0×

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: 19.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 51.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B_m, A\right)\right)}\\ t_1 := \mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)\\ t_2 := \mathsf{fma}\left(A, C \cdot -4, {B_m}^{2}\right)\\ t_3 := C \cdot \left(4 \cdot A\right)\\ t_4 := \frac{-\sqrt{\left(\sqrt{{B_m}^{2} + {\left(A - C\right)}^{2}} - \left(A + C\right)\right) \cdot \left(2 \cdot \left(F \cdot \left(t_3 - {B_m}^{2}\right)\right)\right)}}{{B_m}^{2} - t_3}\\ t_5 := 2 \cdot t_2\\ \mathbf{if}\;t_4 \leq -2 \cdot 10^{-218}:\\ \;\;\;\;\frac{-t_0 \cdot \sqrt{t_5}}{t_2}\\ \mathbf{elif}\;t_4 \leq 0:\\ \;\;\;\;\frac{-\sqrt{\left(F \cdot t_1\right) \cdot \left(2 \cdot \left(A + \left(A + -0.5 \cdot \frac{{A}^{2} + \left({B_m}^{2} - {\left(-A\right)}^{2}\right)}{C}\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;t_4 \leq \infty:\\ \;\;\;\;\frac{\sqrt{t_5 \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B_m, A - C\right)\right)} \cdot \left(-\sqrt{F}\right)}{t_2}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(-\frac{\sqrt{2}}{B_m}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (sqrt (* F (- A (hypot B_m A)))))
        (t_1 (fma B_m B_m (* A (* C -4.0))))
        (t_2 (fma A (* C -4.0) (pow B_m 2.0)))
        (t_3 (* C (* 4.0 A)))
        (t_4
         (/
          (-
           (sqrt
            (*
             (- (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))) (+ A C))
             (* 2.0 (* F (- t_3 (pow B_m 2.0)))))))
          (- (pow B_m 2.0) t_3)))
        (t_5 (* 2.0 t_2)))
   (if (<= t_4 -2e-218)
     (/ (- (* t_0 (sqrt t_5))) t_2)
     (if (<= t_4 0.0)
       (/
        (-
         (sqrt
          (*
           (* F t_1)
           (*
            2.0
            (+
             A
             (+
              A
              (*
               -0.5
               (/ (+ (pow A 2.0) (- (pow B_m 2.0) (pow (- A) 2.0))) C))))))))
        t_1)
       (if (<= t_4 INFINITY)
         (/
          (* (sqrt (* t_5 (- (+ A C) (hypot B_m (- A C))))) (- (sqrt F)))
          t_2)
         (* t_0 (- (/ (sqrt 2.0) B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = sqrt((F * (A - hypot(B_m, A))));
	double t_1 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_2 = fma(A, (C * -4.0), pow(B_m, 2.0));
	double t_3 = C * (4.0 * A);
	double t_4 = -sqrt(((sqrt((pow(B_m, 2.0) + pow((A - C), 2.0))) - (A + C)) * (2.0 * (F * (t_3 - pow(B_m, 2.0)))))) / (pow(B_m, 2.0) - t_3);
	double t_5 = 2.0 * t_2;
	double tmp;
	if (t_4 <= -2e-218) {
		tmp = -(t_0 * sqrt(t_5)) / t_2;
	} else if (t_4 <= 0.0) {
		tmp = -sqrt(((F * t_1) * (2.0 * (A + (A + (-0.5 * ((pow(A, 2.0) + (pow(B_m, 2.0) - pow(-A, 2.0))) / C))))))) / t_1;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = (sqrt((t_5 * ((A + C) - hypot(B_m, (A - C))))) * -sqrt(F)) / t_2;
	} else {
		tmp = t_0 * -(sqrt(2.0) / B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = sqrt(Float64(F * Float64(A - hypot(B_m, A))))
	t_1 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_2 = fma(A, Float64(C * -4.0), (B_m ^ 2.0))
	t_3 = Float64(C * Float64(4.0 * A))
	t_4 = Float64(Float64(-sqrt(Float64(Float64(sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0))) - Float64(A + C)) * Float64(2.0 * Float64(F * Float64(t_3 - (B_m ^ 2.0))))))) / Float64((B_m ^ 2.0) - t_3))
	t_5 = Float64(2.0 * t_2)
	tmp = 0.0
	if (t_4 <= -2e-218)
		tmp = Float64(Float64(-Float64(t_0 * sqrt(t_5))) / t_2);
	elseif (t_4 <= 0.0)
		tmp = Float64(Float64(-sqrt(Float64(Float64(F * t_1) * Float64(2.0 * Float64(A + Float64(A + Float64(-0.5 * Float64(Float64((A ^ 2.0) + Float64((B_m ^ 2.0) - (Float64(-A) ^ 2.0))) / C)))))))) / t_1);
	elseif (t_4 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(t_5 * Float64(Float64(A + C) - hypot(B_m, Float64(A - C))))) * Float64(-sqrt(F))) / t_2);
	else
		tmp = Float64(t_0 * Float64(-Float64(sqrt(2.0) / B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[Sqrt[N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $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[(A * N[(C * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(C * N[(4.0 * A), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[((-N[Sqrt[N[(N[(N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(A + C), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(F * N[(t$95$3 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(2.0 * t$95$2), $MachinePrecision]}, If[LessEqual[t$95$4, -2e-218], N[((-N[(t$95$0 * N[Sqrt[t$95$5], $MachinePrecision]), $MachinePrecision]) / t$95$2), $MachinePrecision], If[LessEqual[t$95$4, 0.0], N[((-N[Sqrt[N[(N[(F * t$95$1), $MachinePrecision] * N[(2.0 * N[(A + N[(A + N[(-0.5 * N[(N[(N[Power[A, 2.0], $MachinePrecision] + N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[Power[(-A), 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[(N[Sqrt[N[(t$95$5 * N[(N[(A + C), $MachinePrecision] - N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision] / t$95$2), $MachinePrecision], N[(t$95$0 * (-N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision])), $MachinePrecision]]]]]]]]]]

Derivation

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

Alternative 2: 51.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B_m, A\right)\right)}\\ t_1 := \mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)\\ t_2 := \mathsf{fma}\left(A, C \cdot -4, {B_m}^{2}\right)\\ t_3 := C \cdot \left(4 \cdot A\right)\\ t_4 := \frac{-\sqrt{\left(\sqrt{{B_m}^{2} + {\left(A - C\right)}^{2}} - \left(A + C\right)\right) \cdot \left(2 \cdot \left(F \cdot \left(t_3 - {B_m}^{2}\right)\right)\right)}}{{B_m}^{2} - t_3}\\ \mathbf{if}\;t_4 \leq -2 \cdot 10^{-218}:\\ \;\;\;\;\frac{-t_0 \cdot \sqrt{2 \cdot t_2}}{t_2}\\ \mathbf{elif}\;t_4 \leq 0:\\ \;\;\;\;\frac{-\sqrt{\left(F \cdot t_1\right) \cdot \left(2 \cdot \left(A + \left(A + -0.5 \cdot \frac{{A}^{2} + \left({B_m}^{2} - {\left(-A\right)}^{2}\right)}{C}\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;t_4 \leq \infty:\\ \;\;\;\;\sqrt{t_1 \cdot \left(F \cdot \left(2 \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B_m, A - C\right)\right)\right)\right)} \cdot \frac{-1}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(-\frac{\sqrt{2}}{B_m}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (sqrt (* F (- A (hypot B_m A)))))
        (t_1 (fma B_m B_m (* A (* C -4.0))))
        (t_2 (fma A (* C -4.0) (pow B_m 2.0)))
        (t_3 (* C (* 4.0 A)))
        (t_4
         (/
          (-
           (sqrt
            (*
             (- (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))) (+ A C))
             (* 2.0 (* F (- t_3 (pow B_m 2.0)))))))
          (- (pow B_m 2.0) t_3))))
   (if (<= t_4 -2e-218)
     (/ (- (* t_0 (sqrt (* 2.0 t_2)))) t_2)
     (if (<= t_4 0.0)
       (/
        (-
         (sqrt
          (*
           (* F t_1)
           (*
            2.0
            (+
             A
             (+
              A
              (*
               -0.5
               (/ (+ (pow A 2.0) (- (pow B_m 2.0) (pow (- A) 2.0))) C))))))))
        t_1)
       (if (<= t_4 INFINITY)
         (*
          (sqrt (* t_1 (* F (* 2.0 (- (+ A C) (hypot B_m (- A C)))))))
          (/ -1.0 t_1))
         (* t_0 (- (/ (sqrt 2.0) B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = sqrt((F * (A - hypot(B_m, A))));
	double t_1 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_2 = fma(A, (C * -4.0), pow(B_m, 2.0));
	double t_3 = C * (4.0 * A);
	double t_4 = -sqrt(((sqrt((pow(B_m, 2.0) + pow((A - C), 2.0))) - (A + C)) * (2.0 * (F * (t_3 - pow(B_m, 2.0)))))) / (pow(B_m, 2.0) - t_3);
	double tmp;
	if (t_4 <= -2e-218) {
		tmp = -(t_0 * sqrt((2.0 * t_2))) / t_2;
	} else if (t_4 <= 0.0) {
		tmp = -sqrt(((F * t_1) * (2.0 * (A + (A + (-0.5 * ((pow(A, 2.0) + (pow(B_m, 2.0) - pow(-A, 2.0))) / C))))))) / t_1;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt((t_1 * (F * (2.0 * ((A + C) - hypot(B_m, (A - C))))))) * (-1.0 / t_1);
	} else {
		tmp = t_0 * -(sqrt(2.0) / B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = sqrt(Float64(F * Float64(A - hypot(B_m, A))))
	t_1 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_2 = fma(A, Float64(C * -4.0), (B_m ^ 2.0))
	t_3 = Float64(C * Float64(4.0 * A))
	t_4 = Float64(Float64(-sqrt(Float64(Float64(sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0))) - Float64(A + C)) * Float64(2.0 * Float64(F * Float64(t_3 - (B_m ^ 2.0))))))) / Float64((B_m ^ 2.0) - t_3))
	tmp = 0.0
	if (t_4 <= -2e-218)
		tmp = Float64(Float64(-Float64(t_0 * sqrt(Float64(2.0 * t_2)))) / t_2);
	elseif (t_4 <= 0.0)
		tmp = Float64(Float64(-sqrt(Float64(Float64(F * t_1) * Float64(2.0 * Float64(A + Float64(A + Float64(-0.5 * Float64(Float64((A ^ 2.0) + Float64((B_m ^ 2.0) - (Float64(-A) ^ 2.0))) / C)))))))) / t_1);
	elseif (t_4 <= Inf)
		tmp = Float64(sqrt(Float64(t_1 * Float64(F * Float64(2.0 * Float64(Float64(A + C) - hypot(B_m, Float64(A - C))))))) * Float64(-1.0 / t_1));
	else
		tmp = Float64(t_0 * Float64(-Float64(sqrt(2.0) / B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[Sqrt[N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $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[(A * N[(C * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(C * N[(4.0 * A), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[((-N[Sqrt[N[(N[(N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(A + C), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(F * N[(t$95$3 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -2e-218], N[((-N[(t$95$0 * N[Sqrt[N[(2.0 * t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]) / t$95$2), $MachinePrecision], If[LessEqual[t$95$4, 0.0], N[((-N[Sqrt[N[(N[(F * t$95$1), $MachinePrecision] * N[(2.0 * N[(A + N[(A + N[(-0.5 * N[(N[(N[Power[A, 2.0], $MachinePrecision] + N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[Power[(-A), 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[Sqrt[N[(t$95$1 * N[(F * N[(2.0 * N[(N[(A + C), $MachinePrecision] - N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / t$95$1), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * (-N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision])), $MachinePrecision]]]]]]]]]

Derivation

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

Alternative 3: 46.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := {B_m}^{2} - C \cdot \left(4 \cdot A\right)\\ t_2 := -\frac{\sqrt{2}}{B_m}\\ \mathbf{if}\;{B_m}^{2} \leq 2 \cdot 10^{-260}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, {B_m}^{2}\right)}\\ \mathbf{elif}\;{B_m}^{2} \leq 2 \cdot 10^{-81}:\\ \;\;\;\;\frac{-\sqrt{\left(F \cdot t_0\right) \cdot \left(2 \cdot \left(A + \left(A + -0.5 \cdot \frac{{A}^{2} + \left({B_m}^{2} - {\left(-A\right)}^{2}\right)}{C}\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;{B_m}^{2} \leq 10^{+81}:\\ \;\;\;\;\frac{1}{\frac{t_1}{-\sqrt{\left(t_1 \cdot \left(2 \cdot F\right)\right) \cdot \left(A + \left(C - \mathsf{hypot}\left(B_m, A - C\right)\right)\right)}}}\\ \mathbf{elif}\;{B_m}^{2} \leq 5 \cdot 10^{+145}:\\ \;\;\;\;t_2 \cdot \sqrt{F \cdot \left(-0.5 \cdot \frac{{B_m}^{2}}{C}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B_m, A\right)\right)} \cdot t_2\\ \end{array} \end{array} \]

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))))
        (t_1 (- (pow B_m 2.0) (* C (* 4.0 A))))
        (t_2 (- (/ (sqrt 2.0) B_m))))
   (if (<= (pow B_m 2.0) 2e-260)
     (/
      (- (sqrt (* -8.0 (* (* A C) (* F (+ A A))))))
      (fma A (* C -4.0) (pow B_m 2.0)))
     (if (<= (pow B_m 2.0) 2e-81)
       (/
        (-
         (sqrt
          (*
           (* F t_0)
           (*
            2.0
            (+
             A
             (+
              A
              (*
               -0.5
               (/ (+ (pow A 2.0) (- (pow B_m 2.0) (pow (- A) 2.0))) C))))))))
        t_0)
       (if (<= (pow B_m 2.0) 1e+81)
         (/
          1.0
          (/
           t_1
           (- (sqrt (* (* t_1 (* 2.0 F)) (+ A (- C (hypot B_m (- A C)))))))))
         (if (<= (pow B_m 2.0) 5e+145)
           (* t_2 (sqrt (* F (* -0.5 (/ (pow B_m 2.0) C)))))
           (* (sqrt (* F (- A (hypot B_m A)))) t_2)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_1 = pow(B_m, 2.0) - (C * (4.0 * A));
	double t_2 = -(sqrt(2.0) / B_m);
	double tmp;
	if (pow(B_m, 2.0) <= 2e-260) {
		tmp = -sqrt((-8.0 * ((A * C) * (F * (A + A))))) / fma(A, (C * -4.0), pow(B_m, 2.0));
	} else if (pow(B_m, 2.0) <= 2e-81) {
		tmp = -sqrt(((F * t_0) * (2.0 * (A + (A + (-0.5 * ((pow(A, 2.0) + (pow(B_m, 2.0) - pow(-A, 2.0))) / C))))))) / t_0;
	} else if (pow(B_m, 2.0) <= 1e+81) {
		tmp = 1.0 / (t_1 / -sqrt(((t_1 * (2.0 * F)) * (A + (C - hypot(B_m, (A - C)))))));
	} else if (pow(B_m, 2.0) <= 5e+145) {
		tmp = t_2 * sqrt((F * (-0.5 * (pow(B_m, 2.0) / C))));
	} else {
		tmp = sqrt((F * (A - hypot(B_m, A)))) * t_2;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_1 = Float64((B_m ^ 2.0) - Float64(C * Float64(4.0 * A)))
	t_2 = Float64(-Float64(sqrt(2.0) / B_m))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-260)
		tmp = Float64(Float64(-sqrt(Float64(-8.0 * Float64(Float64(A * C) * Float64(F * Float64(A + A)))))) / fma(A, Float64(C * -4.0), (B_m ^ 2.0)));
	elseif ((B_m ^ 2.0) <= 2e-81)
		tmp = Float64(Float64(-sqrt(Float64(Float64(F * t_0) * Float64(2.0 * Float64(A + Float64(A + Float64(-0.5 * Float64(Float64((A ^ 2.0) + Float64((B_m ^ 2.0) - (Float64(-A) ^ 2.0))) / C)))))))) / t_0);
	elseif ((B_m ^ 2.0) <= 1e+81)
		tmp = Float64(1.0 / Float64(t_1 / Float64(-sqrt(Float64(Float64(t_1 * Float64(2.0 * F)) * Float64(A + Float64(C - hypot(B_m, Float64(A - C)))))))));
	elseif ((B_m ^ 2.0) <= 5e+145)
		tmp = Float64(t_2 * sqrt(Float64(F * Float64(-0.5 * Float64((B_m ^ 2.0) / C)))));
	else
		tmp = Float64(sqrt(Float64(F * Float64(A - hypot(B_m, A)))) * t_2);
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(C * N[(4.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = (-N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision])}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-260], N[((-N[Sqrt[N[(-8.0 * N[(N[(A * C), $MachinePrecision] * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(A * N[(C * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-81], N[((-N[Sqrt[N[(N[(F * t$95$0), $MachinePrecision] * N[(2.0 * N[(A + N[(A + N[(-0.5 * N[(N[(N[Power[A, 2.0], $MachinePrecision] + N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[Power[(-A), 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+81], N[(1.0 / N[(t$95$1 / (-N[Sqrt[N[(N[(t$95$1 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision] * N[(A + N[(C - N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+145], N[(t$95$2 * N[Sqrt[N[(F * N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision]]]]]]]]

Derivation

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

Alternative 4: 43.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := F \cdot \left(2 \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B_m, A - C\right)\right)\right)\\ t_1 := F \cdot \left(A + A\right)\\ t_2 := \mathsf{fma}\left(A, C \cdot -4, {B_m}^{2}\right)\\ t_3 := \mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)\\ t_4 := -\frac{\sqrt{2}}{B_m}\\ \mathbf{if}\;{B_m}^{2} \leq 2 \cdot 10^{-260}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(\left(A \cdot C\right) \cdot t_1\right)}}{t_2}\\ \mathbf{elif}\;{B_m}^{2} \leq 5 \cdot 10^{-161}:\\ \;\;\;\;\frac{\left|B_m\right| \cdot \left(-\sqrt{t_0}\right)}{t_3}\\ \mathbf{elif}\;{B_m}^{2} \leq 2 \cdot 10^{-81}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(A \cdot \left(C \cdot t_1\right)\right)}}{t_2}\\ \mathbf{elif}\;{B_m}^{2} \leq 10^{+81}:\\ \;\;\;\;\frac{-\sqrt{t_3 \cdot t_0}}{t_3}\\ \mathbf{elif}\;{B_m}^{2} \leq 5 \cdot 10^{+145}:\\ \;\;\;\;t_4 \cdot \sqrt{F \cdot \left(-0.5 \cdot \frac{{B_m}^{2}}{C}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B_m, A\right)\right)} \cdot t_4\\ \end{array} \end{array} \]

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 (* F (* 2.0 (- (+ A C) (hypot B_m (- A C))))))
        (t_1 (* F (+ A A)))
        (t_2 (fma A (* C -4.0) (pow B_m 2.0)))
        (t_3 (fma B_m B_m (* A (* C -4.0))))
        (t_4 (- (/ (sqrt 2.0) B_m))))
   (if (<= (pow B_m 2.0) 2e-260)
     (/ (- (sqrt (* -8.0 (* (* A C) t_1)))) t_2)
     (if (<= (pow B_m 2.0) 5e-161)
       (/ (* (fabs B_m) (- (sqrt t_0))) t_3)
       (if (<= (pow B_m 2.0) 2e-81)
         (/ (- (sqrt (* -8.0 (* A (* C t_1))))) t_2)
         (if (<= (pow B_m 2.0) 1e+81)
           (/ (- (sqrt (* t_3 t_0))) t_3)
           (if (<= (pow B_m 2.0) 5e+145)
             (* t_4 (sqrt (* F (* -0.5 (/ (pow B_m 2.0) C)))))
             (* (sqrt (* F (- A (hypot B_m A)))) 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 = F * (2.0 * ((A + C) - hypot(B_m, (A - C))));
	double t_1 = F * (A + A);
	double t_2 = fma(A, (C * -4.0), pow(B_m, 2.0));
	double t_3 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_4 = -(sqrt(2.0) / B_m);
	double tmp;
	if (pow(B_m, 2.0) <= 2e-260) {
		tmp = -sqrt((-8.0 * ((A * C) * t_1))) / t_2;
	} else if (pow(B_m, 2.0) <= 5e-161) {
		tmp = (fabs(B_m) * -sqrt(t_0)) / t_3;
	} else if (pow(B_m, 2.0) <= 2e-81) {
		tmp = -sqrt((-8.0 * (A * (C * t_1)))) / t_2;
	} else if (pow(B_m, 2.0) <= 1e+81) {
		tmp = -sqrt((t_3 * t_0)) / t_3;
	} else if (pow(B_m, 2.0) <= 5e+145) {
		tmp = t_4 * sqrt((F * (-0.5 * (pow(B_m, 2.0) / C))));
	} else {
		tmp = sqrt((F * (A - hypot(B_m, A)))) * 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(F * Float64(2.0 * Float64(Float64(A + C) - hypot(B_m, Float64(A - C)))))
	t_1 = Float64(F * Float64(A + A))
	t_2 = fma(A, Float64(C * -4.0), (B_m ^ 2.0))
	t_3 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_4 = Float64(-Float64(sqrt(2.0) / B_m))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-260)
		tmp = Float64(Float64(-sqrt(Float64(-8.0 * Float64(Float64(A * C) * t_1)))) / t_2);
	elseif ((B_m ^ 2.0) <= 5e-161)
		tmp = Float64(Float64(abs(B_m) * Float64(-sqrt(t_0))) / t_3);
	elseif ((B_m ^ 2.0) <= 2e-81)
		tmp = Float64(Float64(-sqrt(Float64(-8.0 * Float64(A * Float64(C * t_1))))) / t_2);
	elseif ((B_m ^ 2.0) <= 1e+81)
		tmp = Float64(Float64(-sqrt(Float64(t_3 * t_0))) / t_3);
	elseif ((B_m ^ 2.0) <= 5e+145)
		tmp = Float64(t_4 * sqrt(Float64(F * Float64(-0.5 * Float64((B_m ^ 2.0) / C)))));
	else
		tmp = Float64(sqrt(Float64(F * Float64(A - hypot(B_m, A)))) * 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[(F * N[(2.0 * N[(N[(A + C), $MachinePrecision] - N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(A * N[(C * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = (-N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision])}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-260], N[((-N[Sqrt[N[(-8.0 * N[(N[(A * C), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-161], N[(N[(N[Abs[B$95$m], $MachinePrecision] * (-N[Sqrt[t$95$0], $MachinePrecision])), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-81], N[((-N[Sqrt[N[(-8.0 * N[(A * N[(C * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+81], N[((-N[Sqrt[N[(t$95$3 * t$95$0), $MachinePrecision]], $MachinePrecision]) / t$95$3), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+145], N[(t$95$4 * N[Sqrt[N[(F * N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$4), $MachinePrecision]]]]]]]]]]]

Derivation

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

Alternative 5: 46.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := -\frac{\sqrt{2}}{B_m}\\ \mathbf{if}\;{B_m}^{2} \leq 4 \cdot 10^{-190}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, {B_m}^{2}\right)}\\ \mathbf{elif}\;{B_m}^{2} \leq 10^{+81}:\\ \;\;\;\;\frac{-\sqrt{\left(F \cdot t_0\right) \cdot \left(2 \cdot \left(A + \left(C - \mathsf{hypot}\left(B_m, A - C\right)\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;{B_m}^{2} \leq 5 \cdot 10^{+145}:\\ \;\;\;\;t_1 \cdot \sqrt{F \cdot \left(-0.5 \cdot \frac{{B_m}^{2}}{C}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B_m, A\right)\right)} \cdot t_1\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))) (t_1 (- (/ (sqrt 2.0) B_m))))
   (if (<= (pow B_m 2.0) 4e-190)
     (/
      (- (sqrt (* -8.0 (* (* A C) (* F (+ A A))))))
      (fma A (* C -4.0) (pow B_m 2.0)))
     (if (<= (pow B_m 2.0) 1e+81)
       (/ (- (sqrt (* (* F t_0) (* 2.0 (+ A (- C (hypot B_m (- A C)))))))) t_0)
       (if (<= (pow B_m 2.0) 5e+145)
         (* t_1 (sqrt (* F (* -0.5 (/ (pow B_m 2.0) C)))))
         (* (sqrt (* F (- A (hypot B_m A)))) 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 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_1 = -(sqrt(2.0) / B_m);
	double tmp;
	if (pow(B_m, 2.0) <= 4e-190) {
		tmp = -sqrt((-8.0 * ((A * C) * (F * (A + A))))) / fma(A, (C * -4.0), pow(B_m, 2.0));
	} else if (pow(B_m, 2.0) <= 1e+81) {
		tmp = -sqrt(((F * t_0) * (2.0 * (A + (C - hypot(B_m, (A - C))))))) / t_0;
	} else if (pow(B_m, 2.0) <= 5e+145) {
		tmp = t_1 * sqrt((F * (-0.5 * (pow(B_m, 2.0) / C))));
	} else {
		tmp = sqrt((F * (A - hypot(B_m, A)))) * 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 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_1 = Float64(-Float64(sqrt(2.0) / B_m))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 4e-190)
		tmp = Float64(Float64(-sqrt(Float64(-8.0 * Float64(Float64(A * C) * Float64(F * Float64(A + A)))))) / fma(A, Float64(C * -4.0), (B_m ^ 2.0)));
	elseif ((B_m ^ 2.0) <= 1e+81)
		tmp = Float64(Float64(-sqrt(Float64(Float64(F * t_0) * Float64(2.0 * Float64(A + Float64(C - hypot(B_m, Float64(A - C)))))))) / t_0);
	elseif ((B_m ^ 2.0) <= 5e+145)
		tmp = Float64(t_1 * sqrt(Float64(F * Float64(-0.5 * Float64((B_m ^ 2.0) / C)))));
	else
		tmp = Float64(sqrt(Float64(F * Float64(A - hypot(B_m, A)))) * 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[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision])}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e-190], N[((-N[Sqrt[N[(-8.0 * N[(N[(A * C), $MachinePrecision] * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(A * N[(C * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+81], N[((-N[Sqrt[N[(N[(F * t$95$0), $MachinePrecision] * N[(2.0 * N[(A + N[(C - N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+145], N[(t$95$1 * N[Sqrt[N[(F * N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]

Derivation

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

Alternative 6: 44.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := -\frac{\sqrt{2}}{B_m}\\ \mathbf{if}\;{B_m}^{2} \leq 2 \cdot 10^{-260}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, {B_m}^{2}\right)}\\ \mathbf{elif}\;{B_m}^{2} \leq 5 \cdot 10^{-161}:\\ \;\;\;\;\frac{\left|B_m\right| \cdot \left(-\sqrt{F \cdot \left(2 \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B_m, A - C\right)\right)\right)}\right)}{\mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B_m}^{2} \leq 5 \cdot 10^{-38}:\\ \;\;\;\;t_0 \cdot \sqrt{-0.5 \cdot \frac{{B_m}^{2}}{\frac{C}{F}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B_m, A\right)\right)} \cdot t_0\\ \end{array} \end{array} \]

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) B_m))))
   (if (<= (pow B_m 2.0) 2e-260)
     (/
      (- (sqrt (* -8.0 (* (* A C) (* F (+ A A))))))
      (fma A (* C -4.0) (pow B_m 2.0)))
     (if (<= (pow B_m 2.0) 5e-161)
       (/
        (* (fabs B_m) (- (sqrt (* F (* 2.0 (- (+ A C) (hypot B_m (- A C))))))))
        (fma B_m B_m (* A (* C -4.0))))
       (if (<= (pow B_m 2.0) 5e-38)
         (* t_0 (sqrt (* -0.5 (/ (pow B_m 2.0) (/ C F)))))
         (* (sqrt (* F (- A (hypot B_m A)))) 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(2.0) / B_m);
	double tmp;
	if (pow(B_m, 2.0) <= 2e-260) {
		tmp = -sqrt((-8.0 * ((A * C) * (F * (A + A))))) / fma(A, (C * -4.0), pow(B_m, 2.0));
	} else if (pow(B_m, 2.0) <= 5e-161) {
		tmp = (fabs(B_m) * -sqrt((F * (2.0 * ((A + C) - hypot(B_m, (A - C))))))) / fma(B_m, B_m, (A * (C * -4.0)));
	} else if (pow(B_m, 2.0) <= 5e-38) {
		tmp = t_0 * sqrt((-0.5 * (pow(B_m, 2.0) / (C / F))));
	} else {
		tmp = sqrt((F * (A - hypot(B_m, A)))) * 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(-Float64(sqrt(2.0) / B_m))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-260)
		tmp = Float64(Float64(-sqrt(Float64(-8.0 * Float64(Float64(A * C) * Float64(F * Float64(A + A)))))) / fma(A, Float64(C * -4.0), (B_m ^ 2.0)));
	elseif ((B_m ^ 2.0) <= 5e-161)
		tmp = Float64(Float64(abs(B_m) * Float64(-sqrt(Float64(F * Float64(2.0 * Float64(Float64(A + C) - hypot(B_m, Float64(A - C)))))))) / fma(B_m, B_m, Float64(A * Float64(C * -4.0))));
	elseif ((B_m ^ 2.0) <= 5e-38)
		tmp = Float64(t_0 * sqrt(Float64(-0.5 * Float64((B_m ^ 2.0) / Float64(C / F)))));
	else
		tmp = Float64(sqrt(Float64(F * Float64(A - hypot(B_m, A)))) * 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[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision])}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-260], N[((-N[Sqrt[N[(-8.0 * N[(N[(A * C), $MachinePrecision] * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(A * N[(C * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-161], N[(N[(N[Abs[B$95$m], $MachinePrecision] * (-N[Sqrt[N[(F * N[(2.0 * N[(N[(A + C), $MachinePrecision] - N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-38], N[(t$95$0 * N[Sqrt[N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / N[(C / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]]]]]

Derivation

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1. Add Preprocessing

Alternative 7: 45.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B_m}^{2} \leq 5 \cdot 10^{-38}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, {B_m}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B_m, A\right)\right)} \cdot \left(-\frac{\sqrt{2}}{B_m}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 5e-38)
   (/
    (- (sqrt (* -8.0 (* (* A C) (* F (+ A A))))))
    (fma A (* C -4.0) (pow B_m 2.0)))
   (* (sqrt (* F (- A (hypot B_m A)))) (- (/ (sqrt 2.0) B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 5e-38) {
		tmp = -sqrt((-8.0 * ((A * C) * (F * (A + A))))) / fma(A, (C * -4.0), pow(B_m, 2.0));
	} else {
		tmp = sqrt((F * (A - hypot(B_m, A)))) * -(sqrt(2.0) / B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e-38)
		tmp = Float64(Float64(-sqrt(Float64(-8.0 * Float64(Float64(A * C) * Float64(F * Float64(A + A)))))) / fma(A, Float64(C * -4.0), (B_m ^ 2.0)));
	else
		tmp = Float64(sqrt(Float64(F * Float64(A - hypot(B_m, A)))) * Float64(-Float64(sqrt(2.0) / B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-38], N[((-N[Sqrt[N[(-8.0 * N[(N[(A * C), $MachinePrecision] * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(A * N[(C * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision])), $MachinePrecision]]

Derivation

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1. Add Preprocessing

Alternative 8: 34.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := -\frac{\sqrt{2}}{B_m}\\ \mathbf{if}\;C \leq 2.25 \cdot 10^{+20}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B_m, A\right)\right)} \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \sqrt{-0.5 \cdot \frac{{B_m}^{2}}{\frac{C}{F}}}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (- (/ (sqrt 2.0) B_m))))
   (if (<= C 2.25e+20)
     (* (sqrt (* F (- A (hypot B_m A)))) t_0)
     (* t_0 (sqrt (* -0.5 (/ (pow B_m 2.0) (/ C 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 = -(sqrt(2.0) / B_m);
	double tmp;
	if (C <= 2.25e+20) {
		tmp = sqrt((F * (A - hypot(B_m, A)))) * t_0;
	} else {
		tmp = t_0 * sqrt((-0.5 * (pow(B_m, 2.0) / (C / F))));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = -(Math.sqrt(2.0) / B_m);
	double tmp;
	if (C <= 2.25e+20) {
		tmp = Math.sqrt((F * (A - Math.hypot(B_m, A)))) * t_0;
	} else {
		tmp = t_0 * Math.sqrt((-0.5 * (Math.pow(B_m, 2.0) / (C / 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.sqrt(2.0) / B_m)
	tmp = 0
	if C <= 2.25e+20:
		tmp = math.sqrt((F * (A - math.hypot(B_m, A)))) * t_0
	else:
		tmp = t_0 * math.sqrt((-0.5 * (math.pow(B_m, 2.0) / (C / F))))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(-Float64(sqrt(2.0) / B_m))
	tmp = 0.0
	if (C <= 2.25e+20)
		tmp = Float64(sqrt(Float64(F * Float64(A - hypot(B_m, A)))) * t_0);
	else
		tmp = Float64(t_0 * sqrt(Float64(-0.5 * Float64((B_m ^ 2.0) / Float64(C / 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 = -(sqrt(2.0) / B_m);
	tmp = 0.0;
	if (C <= 2.25e+20)
		tmp = sqrt((F * (A - hypot(B_m, A)))) * t_0;
	else
		tmp = t_0 * sqrt((-0.5 * ((B_m ^ 2.0) / (C / 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[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision])}, If[LessEqual[C, 2.25e+20], N[(N[Sqrt[N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(t$95$0 * N[Sqrt[N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / N[(C / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

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

Alternative 9: 34.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := -\frac{\sqrt{2}}{B_m}\\ \mathbf{if}\;C \leq 3.8 \cdot 10^{+20}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B_m, A\right)\right)} \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \sqrt{F \cdot \left(-0.5 \cdot \frac{{B_m}^{2}}{C}\right)}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (/ (sqrt 2.0) B_m))))
   (if (<= C 3.8e+20)
     (* (sqrt (* F (- A (hypot B_m A)))) t_0)
     (* t_0 (sqrt (* F (* -0.5 (/ (pow B_m 2.0) C))))))))

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) / B_m);
	double tmp;
	if (C <= 3.8e+20) {
		tmp = sqrt((F * (A - hypot(B_m, A)))) * t_0;
	} else {
		tmp = t_0 * sqrt((F * (-0.5 * (pow(B_m, 2.0) / C))));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = -(Math.sqrt(2.0) / B_m);
	double tmp;
	if (C <= 3.8e+20) {
		tmp = Math.sqrt((F * (A - Math.hypot(B_m, A)))) * t_0;
	} else {
		tmp = t_0 * Math.sqrt((F * (-0.5 * (Math.pow(B_m, 2.0) / C))));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = -(math.sqrt(2.0) / B_m)
	tmp = 0
	if C <= 3.8e+20:
		tmp = math.sqrt((F * (A - math.hypot(B_m, A)))) * t_0
	else:
		tmp = t_0 * math.sqrt((F * (-0.5 * (math.pow(B_m, 2.0) / C))))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(-Float64(sqrt(2.0) / B_m))
	tmp = 0.0
	if (C <= 3.8e+20)
		tmp = Float64(sqrt(Float64(F * Float64(A - hypot(B_m, A)))) * t_0);
	else
		tmp = Float64(t_0 * sqrt(Float64(F * Float64(-0.5 * Float64((B_m ^ 2.0) / C)))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = -(sqrt(2.0) / B_m);
	tmp = 0.0;
	if (C <= 3.8e+20)
		tmp = sqrt((F * (A - hypot(B_m, A)))) * t_0;
	else
		tmp = t_0 * sqrt((F * (-0.5 * ((B_m ^ 2.0) / C))));
	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[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision])}, If[LessEqual[C, 3.8e+20], N[(N[Sqrt[N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(t$95$0 * N[Sqrt[N[(F * N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

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

Alternative 10: 31.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

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 * (A - Math.hypot(B_m, A)))) * -(Math.sqrt(2.0) / B_m);
}

Python

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

Julia

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

MATLAB

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

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := N[(N[Sqrt[N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision])), $MachinePrecision]

Derivation

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

Alternative 11: 26.8% accurate, 3.0× speedup

Math

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

FPCore

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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

Alternative 12: 1.2% accurate, 5.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ 0.25 \cdot \left(\frac{B_m}{\frac{A}{2}} \cdot \sqrt{\frac{F}{C}}\right) \end{array} \]

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
 (* 0.25 (* (/ B_m (/ A 2.0)) (sqrt (/ F C)))))

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 0.25 * ((B_m / (A / 2.0)) * sqrt((F / C)));
}

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 = 0.25d0 * ((b_m / (a / 2.0d0)) * sqrt((f / c)))
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 0.25 * ((B_m / (A / 2.0)) * Math.sqrt((F / C)));
}

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 0.25 * ((B_m / (A / 2.0)) * math.sqrt((F / C)))

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(0.25 * Float64(Float64(B_m / Float64(A / 2.0)) * sqrt(Float64(F / C))))
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 = 0.25 * ((B_m / (A / 2.0)) * sqrt((F / C)));
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[(0.25 * N[(N[(B$95$m / N[(A / 2.0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(F / C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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

Reproduce

herbie shell --seed 2023343 
(FPCore (A B C F)
  :name "ABCF->ab-angle b"
  :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))))