ABCF->ab-angle b

Percentage Accurate: 19.4% → 45.7%

Time: 32.2s

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: 19.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 45.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, {B_m}^{2}\right)\\ t_1 := \mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)\\ t_2 := \left(4 \cdot A\right) \cdot C\\ t_3 := -\frac{\sqrt{\left(\sqrt{{B_m}^{2} + {\left(A - C\right)}^{2}} - \left(A + C\right)\right) \cdot \left(2 \cdot \left(F \cdot \left(t_2 - {B_m}^{2}\right)\right)\right)}}{{B_m}^{2} - t_2}\\ t_4 := \mathsf{hypot}\left(B_m, A - C\right)\\ \mathbf{if}\;t_3 \leq -1 \cdot 10^{-203}:\\ \;\;\;\;\frac{\left(-\sqrt{t_0}\right) \cdot \sqrt{\left(2 \cdot F\right) \cdot \left(C + \left(A - t_4\right)\right)}}{t_0}\\ \mathbf{elif}\;t_3 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(\left(A + A\right) - -0.5 \cdot \frac{\left({\left(-A\right)}^{2} - {A}^{2}\right) - {B_m}^{2}}{C}\right)\right)\right)} \cdot \frac{1}{-t_0}\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;\frac{-\sqrt{\left(F \cdot t_1\right) \cdot \left(2 \cdot \left(A - \left(t_4 - C\right)\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B_m} \cdot \sqrt{F \cdot \left(C - B_m\right)}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma A (* C -4.0) (pow B_m 2.0)))
        (t_1 (fma B_m B_m (* A (* C -4.0))))
        (t_2 (* (* 4.0 A) C))
        (t_3
         (-
          (/
           (sqrt
            (*
             (- (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))) (+ A C))
             (* 2.0 (* F (- t_2 (pow B_m 2.0))))))
           (- (pow B_m 2.0) t_2))))
        (t_4 (hypot B_m (- A C))))
   (if (<= t_3 -1e-203)
     (/ (* (- (sqrt t_0)) (sqrt (* (* 2.0 F) (+ C (- A t_4))))) t_0)
     (if (<= t_3 0.0)
       (*
        (sqrt
         (*
          2.0
          (*
           t_0
           (*
            F
            (-
             (+ A A)
             (*
              -0.5
              (/ (- (- (pow (- A) 2.0) (pow A 2.0)) (pow B_m 2.0)) C)))))))
        (/ 1.0 (- t_0)))
       (if (<= t_3 INFINITY)
         (/ (- (sqrt (* (* F t_1) (* 2.0 (- A (- t_4 C)))))) t_1)
         (* (/ (- (sqrt 2.0)) B_m) (sqrt (* F (- C B_m)))))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(A, (C * -4.0), pow(B_m, 2.0));
	double t_1 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_2 = (4.0 * A) * C;
	double t_3 = -(sqrt(((sqrt((pow(B_m, 2.0) + pow((A - C), 2.0))) - (A + C)) * (2.0 * (F * (t_2 - pow(B_m, 2.0)))))) / (pow(B_m, 2.0) - t_2));
	double t_4 = hypot(B_m, (A - C));
	double tmp;
	if (t_3 <= -1e-203) {
		tmp = (-sqrt(t_0) * sqrt(((2.0 * F) * (C + (A - t_4))))) / t_0;
	} else if (t_3 <= 0.0) {
		tmp = sqrt((2.0 * (t_0 * (F * ((A + A) - (-0.5 * (((pow(-A, 2.0) - pow(A, 2.0)) - pow(B_m, 2.0)) / C))))))) * (1.0 / -t_0);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = -sqrt(((F * t_1) * (2.0 * (A - (t_4 - C))))) / t_1;
	} else {
		tmp = (-sqrt(2.0) / B_m) * sqrt((F * (C - B_m)));
	}
	return tmp;
}

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = fma(A, Float64(C * -4.0), (B_m ^ 2.0))
	t_1 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_2 = Float64(Float64(4.0 * A) * C)
	t_3 = 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_2 - (B_m ^ 2.0)))))) / Float64((B_m ^ 2.0) - t_2)))
	t_4 = hypot(B_m, Float64(A - C))
	tmp = 0.0
	if (t_3 <= -1e-203)
		tmp = Float64(Float64(Float64(-sqrt(t_0)) * sqrt(Float64(Float64(2.0 * F) * Float64(C + Float64(A - t_4))))) / t_0);
	elseif (t_3 <= 0.0)
		tmp = Float64(sqrt(Float64(2.0 * Float64(t_0 * Float64(F * Float64(Float64(A + A) - Float64(-0.5 * Float64(Float64(Float64((Float64(-A) ^ 2.0) - (A ^ 2.0)) - (B_m ^ 2.0)) / C))))))) * Float64(1.0 / Float64(-t_0)));
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(-sqrt(Float64(Float64(F * t_1) * Float64(2.0 * Float64(A - Float64(t_4 - C)))))) / t_1);
	else
		tmp = Float64(Float64(Float64(-sqrt(2.0)) / B_m) * sqrt(Float64(F * Float64(C - B_m))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(A * N[(C * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$3 = (-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$2 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision])}, Block[{t$95$4 = N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[t$95$3, -1e-203], N[(N[((-N[Sqrt[t$95$0], $MachinePrecision]) * N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(C + N[(A - t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(N[Sqrt[N[(2.0 * N[(t$95$0 * N[(F * N[(N[(A + A), $MachinePrecision] - N[(-0.5 * N[(N[(N[(N[Power[(-A), 2.0], $MachinePrecision] - N[Power[A, 2.0], $MachinePrecision]), $MachinePrecision] - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 / (-t$95$0)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[((-N[Sqrt[N[(N[(F * t$95$1), $MachinePrecision] * N[(2.0 * N[(A - N[(t$95$4 - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], N[(N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision] * N[Sqrt[N[(F * N[(C - B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

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

Alternative 2: 45.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := -\frac{\sqrt{\left(\sqrt{{B_m}^{2} + {\left(A - C\right)}^{2}} - \left(A + C\right)\right) \cdot \left(2 \cdot \left(F \cdot \left(t_0 - {B_m}^{2}\right)\right)\right)}}{{B_m}^{2} - t_0}\\ t_2 := \mathsf{hypot}\left(B_m, A - C\right)\\ t_3 := \mathsf{fma}\left(A, C \cdot -4, {B_m}^{2}\right)\\ t_4 := \mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)\\ t_5 := F \cdot t_4\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{-203}:\\ \;\;\;\;\frac{\left(-\sqrt{2 \cdot t_3}\right) \cdot \sqrt{F \cdot \left(C + \left(A - t_2\right)\right)}}{t_3}\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{-\sqrt{t_5 \cdot \left(2 \cdot \left(A + \left(A - -0.5 \cdot \frac{{\left(-A\right)}^{2} - \left({B_m}^{2} + {A}^{2}\right)}{C}\right)\right)\right)}}{t_4}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\frac{-\sqrt{t_5 \cdot \left(2 \cdot \left(A - \left(t_2 - C\right)\right)\right)}}{t_4}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B_m} \cdot \sqrt{F \cdot \left(C - B_m\right)}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C))
        (t_1
         (-
          (/
           (sqrt
            (*
             (- (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))) (+ A C))
             (* 2.0 (* F (- t_0 (pow B_m 2.0))))))
           (- (pow B_m 2.0) t_0))))
        (t_2 (hypot B_m (- A C)))
        (t_3 (fma A (* C -4.0) (pow B_m 2.0)))
        (t_4 (fma B_m B_m (* A (* C -4.0))))
        (t_5 (* F t_4)))
   (if (<= t_1 -1e-203)
     (/ (* (- (sqrt (* 2.0 t_3))) (sqrt (* F (+ C (- A t_2))))) t_3)
     (if (<= t_1 0.0)
       (/
        (-
         (sqrt
          (*
           t_5
           (*
            2.0
            (+
             A
             (-
              A
              (*
               -0.5
               (/ (- (pow (- A) 2.0) (+ (pow B_m 2.0) (pow A 2.0))) C))))))))
        t_4)
       (if (<= t_1 INFINITY)
         (/ (- (sqrt (* t_5 (* 2.0 (- A (- t_2 C)))))) t_4)
         (* (/ (- (sqrt 2.0)) B_m) (sqrt (* F (- C B_m)))))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = -(sqrt(((sqrt((pow(B_m, 2.0) + pow((A - C), 2.0))) - (A + C)) * (2.0 * (F * (t_0 - pow(B_m, 2.0)))))) / (pow(B_m, 2.0) - t_0));
	double t_2 = hypot(B_m, (A - C));
	double t_3 = fma(A, (C * -4.0), pow(B_m, 2.0));
	double t_4 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_5 = F * t_4;
	double tmp;
	if (t_1 <= -1e-203) {
		tmp = (-sqrt((2.0 * t_3)) * sqrt((F * (C + (A - t_2))))) / t_3;
	} else if (t_1 <= 0.0) {
		tmp = -sqrt((t_5 * (2.0 * (A + (A - (-0.5 * ((pow(-A, 2.0) - (pow(B_m, 2.0) + pow(A, 2.0))) / C))))))) / t_4;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = -sqrt((t_5 * (2.0 * (A - (t_2 - C))))) / t_4;
	} else {
		tmp = (-sqrt(2.0) / B_m) * sqrt((F * (C - B_m)));
	}
	return tmp;
}

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = 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_0 - (B_m ^ 2.0)))))) / Float64((B_m ^ 2.0) - t_0)))
	t_2 = hypot(B_m, Float64(A - C))
	t_3 = fma(A, Float64(C * -4.0), (B_m ^ 2.0))
	t_4 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_5 = Float64(F * t_4)
	tmp = 0.0
	if (t_1 <= -1e-203)
		tmp = Float64(Float64(Float64(-sqrt(Float64(2.0 * t_3))) * sqrt(Float64(F * Float64(C + Float64(A - t_2))))) / t_3);
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(-sqrt(Float64(t_5 * Float64(2.0 * Float64(A + Float64(A - Float64(-0.5 * Float64(Float64((Float64(-A) ^ 2.0) - Float64((B_m ^ 2.0) + (A ^ 2.0))) / C)))))))) / t_4);
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(-sqrt(Float64(t_5 * Float64(2.0 * Float64(A - Float64(t_2 - C)))))) / t_4);
	else
		tmp = Float64(Float64(Float64(-sqrt(2.0)) / B_m) * sqrt(Float64(F * Float64(C - B_m))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = (-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$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision])}, Block[{t$95$2 = N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$3 = N[(A * N[(C * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(F * t$95$4), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-203], N[(N[((-N[Sqrt[N[(2.0 * t$95$3), $MachinePrecision]], $MachinePrecision]) * N[Sqrt[N[(F * N[(C + N[(A - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[((-N[Sqrt[N[(t$95$5 * 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$4), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[((-N[Sqrt[N[(t$95$5 * N[(2.0 * N[(A - N[(t$95$2 - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$4), $MachinePrecision], N[(N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision] * N[Sqrt[N[(F * N[(C - B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

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

Alternative 3: 45.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := -\frac{\sqrt{\left(\sqrt{{B_m}^{2} + {\left(A - C\right)}^{2}} - \left(A + C\right)\right) \cdot \left(2 \cdot \left(F \cdot \left(t_1 - {B_m}^{2}\right)\right)\right)}}{{B_m}^{2} - t_1}\\ t_3 := \mathsf{hypot}\left(B_m, A - C\right)\\ t_4 := \mathsf{fma}\left(A, C \cdot -4, {B_m}^{2}\right)\\ t_5 := F \cdot t_0\\ \mathbf{if}\;t_2 \leq -1 \cdot 10^{-203}:\\ \;\;\;\;\frac{\left(-\sqrt{t_4}\right) \cdot \sqrt{\left(2 \cdot F\right) \cdot \left(C + \left(A - t_3\right)\right)}}{t_4}\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;\frac{-\sqrt{t_5 \cdot \left(2 \cdot \left(A + \left(A - -0.5 \cdot \frac{{\left(-A\right)}^{2} - \left({B_m}^{2} + {A}^{2}\right)}{C}\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;\frac{-\sqrt{t_5 \cdot \left(2 \cdot \left(A - \left(t_3 - C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B_m} \cdot \sqrt{F \cdot \left(C - B_m\right)}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0))))
        (t_1 (* (* 4.0 A) C))
        (t_2
         (-
          (/
           (sqrt
            (*
             (- (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))) (+ A C))
             (* 2.0 (* F (- t_1 (pow B_m 2.0))))))
           (- (pow B_m 2.0) t_1))))
        (t_3 (hypot B_m (- A C)))
        (t_4 (fma A (* C -4.0) (pow B_m 2.0)))
        (t_5 (* F t_0)))
   (if (<= t_2 -1e-203)
     (/ (* (- (sqrt t_4)) (sqrt (* (* 2.0 F) (+ C (- A t_3))))) t_4)
     (if (<= t_2 0.0)
       (/
        (-
         (sqrt
          (*
           t_5
           (*
            2.0
            (+
             A
             (-
              A
              (*
               -0.5
               (/ (- (pow (- A) 2.0) (+ (pow B_m 2.0) (pow A 2.0))) C))))))))
        t_0)
       (if (<= t_2 INFINITY)
         (/ (- (sqrt (* t_5 (* 2.0 (- A (- t_3 C)))))) t_0)
         (* (/ (- (sqrt 2.0)) B_m) (sqrt (* F (- C B_m)))))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_1 = (4.0 * A) * C;
	double t_2 = -(sqrt(((sqrt((pow(B_m, 2.0) + pow((A - C), 2.0))) - (A + C)) * (2.0 * (F * (t_1 - pow(B_m, 2.0)))))) / (pow(B_m, 2.0) - t_1));
	double t_3 = hypot(B_m, (A - C));
	double t_4 = fma(A, (C * -4.0), pow(B_m, 2.0));
	double t_5 = F * t_0;
	double tmp;
	if (t_2 <= -1e-203) {
		tmp = (-sqrt(t_4) * sqrt(((2.0 * F) * (C + (A - t_3))))) / t_4;
	} else if (t_2 <= 0.0) {
		tmp = -sqrt((t_5 * (2.0 * (A + (A - (-0.5 * ((pow(-A, 2.0) - (pow(B_m, 2.0) + pow(A, 2.0))) / C))))))) / t_0;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = -sqrt((t_5 * (2.0 * (A - (t_3 - C))))) / t_0;
	} else {
		tmp = (-sqrt(2.0) / B_m) * sqrt((F * (C - B_m)));
	}
	return tmp;
}

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_1 = Float64(Float64(4.0 * A) * C)
	t_2 = 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_1 - (B_m ^ 2.0)))))) / Float64((B_m ^ 2.0) - t_1)))
	t_3 = hypot(B_m, Float64(A - C))
	t_4 = fma(A, Float64(C * -4.0), (B_m ^ 2.0))
	t_5 = Float64(F * t_0)
	tmp = 0.0
	if (t_2 <= -1e-203)
		tmp = Float64(Float64(Float64(-sqrt(t_4)) * sqrt(Float64(Float64(2.0 * F) * Float64(C + Float64(A - t_3))))) / t_4);
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(-sqrt(Float64(t_5 * Float64(2.0 * Float64(A + Float64(A - Float64(-0.5 * Float64(Float64((Float64(-A) ^ 2.0) - Float64((B_m ^ 2.0) + (A ^ 2.0))) / C)))))))) / t_0);
	elseif (t_2 <= Inf)
		tmp = Float64(Float64(-sqrt(Float64(t_5 * Float64(2.0 * Float64(A - Float64(t_3 - C)))))) / t_0);
	else
		tmp = Float64(Float64(Float64(-sqrt(2.0)) / B_m) * sqrt(Float64(F * Float64(C - B_m))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$2 = (-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$1 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision])}, Block[{t$95$3 = N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$4 = N[(A * N[(C * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(F * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-203], N[(N[((-N[Sqrt[t$95$4], $MachinePrecision]) * N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(C + N[(A - t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[((-N[Sqrt[N[(t$95$5 * 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[t$95$2, Infinity], N[((-N[Sqrt[N[(t$95$5 * N[(2.0 * N[(A - N[(t$95$3 - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision] * N[Sqrt[N[(F * N[(C - B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

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

Alternative 4: 38.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := {B_m}^{2} - 4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;{B_m}^{2} \leq 2 \cdot 10^{+100}:\\ \;\;\;\;\frac{-\sqrt{\left(C + \left(A - \mathsf{hypot}\left(B_m, A - C\right)\right)\right) \cdot \left(F \cdot \left(2 \cdot t_0\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B_m} \cdot \sqrt{F \cdot \left(C - B_m\right)}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (pow B_m 2.0) (* 4.0 (* A C)))))
   (if (<= (pow B_m 2.0) 2e+100)
     (/ (- (sqrt (* (+ C (- A (hypot B_m (- A C)))) (* F (* 2.0 t_0))))) t_0)
     (* (/ (- (sqrt 2.0)) B_m) (sqrt (* F (- C B_m)))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = pow(B_m, 2.0) - (4.0 * (A * C));
	double tmp;
	if (pow(B_m, 2.0) <= 2e+100) {
		tmp = -sqrt(((C + (A - hypot(B_m, (A - C)))) * (F * (2.0 * t_0)))) / t_0;
	} else {
		tmp = (-sqrt(2.0) / B_m) * sqrt((F * (C - B_m)));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double t_0 = Math.pow(B_m, 2.0) - (4.0 * (A * C));
	double tmp;
	if (Math.pow(B_m, 2.0) <= 2e+100) {
		tmp = -Math.sqrt(((C + (A - Math.hypot(B_m, (A - C)))) * (F * (2.0 * t_0)))) / t_0;
	} else {
		tmp = (-Math.sqrt(2.0) / B_m) * Math.sqrt((F * (C - B_m)));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = math.pow(B_m, 2.0) - (4.0 * (A * C))
	tmp = 0
	if math.pow(B_m, 2.0) <= 2e+100:
		tmp = -math.sqrt(((C + (A - math.hypot(B_m, (A - C)))) * (F * (2.0 * t_0)))) / t_0
	else:
		tmp = (-math.sqrt(2.0) / B_m) * math.sqrt((F * (C - B_m)))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64((B_m ^ 2.0) - Float64(4.0 * Float64(A * C)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e+100)
		tmp = Float64(Float64(-sqrt(Float64(Float64(C + Float64(A - hypot(B_m, Float64(A - C)))) * Float64(F * Float64(2.0 * t_0))))) / t_0);
	else
		tmp = Float64(Float64(Float64(-sqrt(2.0)) / B_m) * sqrt(Float64(F * Float64(C - B_m))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = (B_m ^ 2.0) - (4.0 * (A * C));
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 2e+100)
		tmp = -sqrt(((C + (A - hypot(B_m, (A - C)))) * (F * (2.0 * t_0)))) / t_0;
	else
		tmp = (-sqrt(2.0) / B_m) * sqrt((F * (C - B_m)));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+100], N[((-N[Sqrt[N[(N[(C + N[(A - N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(F * N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision] * N[Sqrt[N[(F * N[(C - B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

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

Alternative 5: 38.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B_m \leq 1.4 \cdot 10^{+50}:\\ \;\;\;\;\frac{-\sqrt{\left(F \cdot t_0\right) \cdot \left(2 \cdot \left(A - \left(\mathsf{hypot}\left(B_m, A - C\right) - C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B_m} \cdot \sqrt{F \cdot \left(C - B_m\right)}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= B_m 1.4e+50)
     (/ (- (sqrt (* (* F t_0) (* 2.0 (- A (- (hypot B_m (- A C)) C)))))) t_0)
     (* (/ (- (sqrt 2.0)) B_m) (sqrt (* F (- C B_m)))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (B_m <= 1.4e+50) {
		tmp = -sqrt(((F * t_0) * (2.0 * (A - (hypot(B_m, (A - C)) - C))))) / t_0;
	} else {
		tmp = (-sqrt(2.0) / B_m) * sqrt((F * (C - B_m)));
	}
	return tmp;
}

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if (B_m <= 1.4e+50)
		tmp = Float64(Float64(-sqrt(Float64(Float64(F * t_0) * Float64(2.0 * Float64(A - Float64(hypot(B_m, Float64(A - C)) - C)))))) / t_0);
	else
		tmp = Float64(Float64(Float64(-sqrt(2.0)) / B_m) * sqrt(Float64(F * Float64(C - B_m))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 1.4e+50], N[((-N[Sqrt[N[(N[(F * t$95$0), $MachinePrecision] * N[(2.0 * N[(A - N[(N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision] - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision] * N[Sqrt[N[(F * N[(C - B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

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

Alternative 6: 29.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \frac{-\sqrt{2}}{B_m}\\ t_1 := \mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;C \leq -2.6 \cdot 10^{-7}:\\ \;\;\;\;\frac{-\sqrt{\left(F \cdot t_1\right) \cdot \left(2 \cdot \left(A + \left(2 \cdot C - A\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;C \leq 7 \cdot 10^{+104}:\\ \;\;\;\;t_0 \cdot \sqrt{F \cdot \left(C - \left(B_m + 0.5 \cdot \frac{{C}^{2}}{B_m}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \sqrt{F \cdot \left(-0.5 \cdot \frac{{B_m}^{2}}{C}\right)}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt 2.0)) B_m)) (t_1 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= C -2.6e-7)
     (/ (- (sqrt (* (* F t_1) (* 2.0 (+ A (- (* 2.0 C) A)))))) t_1)
     (if (<= C 7e+104)
       (* t_0 (sqrt (* F (- C (+ B_m (* 0.5 (/ (pow C 2.0) B_m)))))))
       (* t_0 (sqrt (* F (* -0.5 (/ (pow B_m 2.0) C)))))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = -sqrt(2.0) / B_m;
	double t_1 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (C <= -2.6e-7) {
		tmp = -sqrt(((F * t_1) * (2.0 * (A + ((2.0 * C) - A))))) / t_1;
	} else if (C <= 7e+104) {
		tmp = t_0 * sqrt((F * (C - (B_m + (0.5 * (pow(C, 2.0) / B_m))))));
	} else {
		tmp = t_0 * sqrt((F * (-0.5 * (pow(B_m, 2.0) / C))));
	}
	return tmp;
}

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(Float64(-sqrt(2.0)) / B_m)
	t_1 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if (C <= -2.6e-7)
		tmp = Float64(Float64(-sqrt(Float64(Float64(F * t_1) * Float64(2.0 * Float64(A + Float64(Float64(2.0 * C) - A)))))) / t_1);
	elseif (C <= 7e+104)
		tmp = Float64(t_0 * sqrt(Float64(F * Float64(C - Float64(B_m + Float64(0.5 * Float64((C ^ 2.0) / B_m)))))));
	else
		tmp = Float64(t_0 * sqrt(Float64(F * Float64(-0.5 * Float64((B_m ^ 2.0) / C)))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[C, -2.6e-7], N[((-N[Sqrt[N[(N[(F * t$95$1), $MachinePrecision] * N[(2.0 * N[(A + N[(N[(2.0 * C), $MachinePrecision] - A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], If[LessEqual[C, 7e+104], N[(t$95$0 * N[Sqrt[N[(F * N[(C - N[(B$95$m + N[(0.5 * N[(N[Power[C, 2.0], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $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 7: 29.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

B_m = fabs(B);
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 (F <= -5e-310) {
		tmp = (-sqrt(2.0) / B_m) * sqrt((F * (C - B_m)));
	} else {
		tmp = -sqrt(((F * (2.0 * t_0)) * (A + (A + C)))) / t_0;
	}
	return tmp;
}

Fortran

B_m = abs(B)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b_m ** 2.0d0) - (4.0d0 * (a * c))
    if (f <= (-5d-310)) then
        tmp = (-sqrt(2.0d0) / b_m) * sqrt((f * (c - b_m)))
    else
        tmp = -sqrt(((f * (2.0d0 * t_0)) * (a + (a + c)))) / t_0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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1. Add Preprocessing

Alternative 8: 27.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \frac{-\sqrt{2}}{B_m}\\ \mathbf{if}\;C \leq 3.9 \cdot 10^{+104}:\\ \;\;\;\;t_0 \cdot \sqrt{F \cdot \left(C - B_m\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \sqrt{F \cdot \left(-0.5 \cdot \frac{{B_m}^{2}}{C}\right)}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt 2.0)) B_m)))
   (if (<= C 3.9e+104)
     (* t_0 (sqrt (* F (- C B_m))))
     (* t_0 (sqrt (* F (* -0.5 (/ (pow B_m 2.0) C))))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = -sqrt(2.0) / B_m;
	double tmp;
	if (C <= 3.9e+104) {
		tmp = t_0 * sqrt((F * (C - B_m)));
	} else {
		tmp = t_0 * sqrt((F * (-0.5 * (pow(B_m, 2.0) / C))));
	}
	return tmp;
}

Fortran

B_m = abs(B)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -sqrt(2.0d0) / b_m
    if (c <= 3.9d+104) then
        tmp = t_0 * sqrt((f * (c - b_m)))
    else
        tmp = t_0 * sqrt((f * ((-0.5d0) * ((b_m ** 2.0d0) / c))))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double t_0 = -Math.sqrt(2.0) / B_m;
	double tmp;
	if (C <= 3.9e+104) {
		tmp = t_0 * Math.sqrt((F * (C - B_m)));
	} else {
		tmp = t_0 * Math.sqrt((F * (-0.5 * (Math.pow(B_m, 2.0) / C))));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = -math.sqrt(2.0) / B_m
	tmp = 0
	if C <= 3.9e+104:
		tmp = t_0 * math.sqrt((F * (C - B_m)))
	else:
		tmp = t_0 * math.sqrt((F * (-0.5 * (math.pow(B_m, 2.0) / C))))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(Float64(-sqrt(2.0)) / B_m)
	tmp = 0.0
	if (C <= 3.9e+104)
		tmp = Float64(t_0 * sqrt(Float64(F * Float64(C - B_m))));
	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);
function tmp_2 = code(A, B_m, C, F)
	t_0 = -sqrt(2.0) / B_m;
	tmp = 0.0;
	if (C <= 3.9e+104)
		tmp = t_0 * sqrt((F * (C - B_m)));
	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]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision]}, If[LessEqual[C, 3.9e+104], N[(t$95$0 * N[Sqrt[N[(F * N[(C - B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $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 9: 27.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-\sqrt{2}}{B_m} \cdot \sqrt{F \cdot \left(C - B_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A + C\right)\right)\right)\right)}}{\mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F -5e-310)
   (* (/ (- (sqrt 2.0)) B_m) (sqrt (* F (- C B_m))))
   (/
    (- (sqrt (* -8.0 (* A (* C (* F (+ A C)))))))
    (fma B_m B_m (* A (* C -4.0))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= -5e-310) {
		tmp = (-sqrt(2.0) / B_m) * sqrt((F * (C - B_m)));
	} else {
		tmp = -sqrt((-8.0 * (A * (C * (F * (A + C)))))) / fma(B_m, B_m, (A * (C * -4.0)));
	}
	return tmp;
}

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= -5e-310)
		tmp = Float64(Float64(Float64(-sqrt(2.0)) / B_m) * sqrt(Float64(F * Float64(C - B_m))));
	else
		tmp = Float64(Float64(-sqrt(Float64(-8.0 * Float64(A * Float64(C * Float64(F * Float64(A + C))))))) / fma(B_m, B_m, Float64(A * Float64(C * -4.0))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, -5e-310], N[(N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision] * N[Sqrt[N[(F * N[(C - B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[N[(-8.0 * N[(A * N[(C * N[(F * N[(A + C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

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

Alternative 10: 1.3% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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

Alternative 11: 4.9% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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

Alternative 12: 25.8% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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

Reproduce

herbie shell --seed 2024006 
(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))))