Toniolo and Linder, Equation (7)

Percentage Accurate: 33.5% → 84.6%

Time: 20.1s

Alternatives: 10

Speedup: 225.0×

Specification

Math

\[\begin{array}{l} \\ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (/
  (* (sqrt 2.0) t)
  (sqrt (- (* (/ (+ x 1.0) (- x 1.0)) (+ (* l l) (* 2.0 (* t t)))) (* l l)))))

C

double code(double x, double l, double t) {
	return (sqrt(2.0) * t) / sqrt(((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l)));
}

Fortran

real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    code = (sqrt(2.0d0) * t) / sqrt(((((x + 1.0d0) / (x - 1.0d0)) * ((l * l) + (2.0d0 * (t * t)))) - (l * l)))
end function

Java

public static double code(double x, double l, double t) {
	return (Math.sqrt(2.0) * t) / Math.sqrt(((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l)));
}

Python

def code(x, l, t):
	return (math.sqrt(2.0) * t) / math.sqrt(((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l)))

Julia

function code(x, l, t)
	return Float64(Float64(sqrt(2.0) * t) / sqrt(Float64(Float64(Float64(Float64(x + 1.0) / Float64(x - 1.0)) * Float64(Float64(l * l) + Float64(2.0 * Float64(t * t)))) - Float64(l * l))))
end

MATLAB

function tmp = code(x, l, t)
	tmp = (sqrt(2.0) * t) / sqrt(((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l)));
end

Wolfram

code[x_, l_, t_] := N[(N[(N[Sqrt[2.0], $MachinePrecision] * t), $MachinePrecision] / N[Sqrt[N[(N[(N[(N[(x + 1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] + N[(2.0 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(l * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 33.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (/
  (* (sqrt 2.0) t)
  (sqrt (- (* (/ (+ x 1.0) (- x 1.0)) (+ (* l l) (* 2.0 (* t t)))) (* l l)))))

C

double code(double x, double l, double t) {
	return (sqrt(2.0) * t) / sqrt(((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l)));
}

Fortran

real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    code = (sqrt(2.0d0) * t) / sqrt(((((x + 1.0d0) / (x - 1.0d0)) * ((l * l) + (2.0d0 * (t * t)))) - (l * l)))
end function

Java

public static double code(double x, double l, double t) {
	return (Math.sqrt(2.0) * t) / Math.sqrt(((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l)));
}

Python

def code(x, l, t):
	return (math.sqrt(2.0) * t) / math.sqrt(((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l)))

Julia

function code(x, l, t)
	return Float64(Float64(sqrt(2.0) * t) / sqrt(Float64(Float64(Float64(Float64(x + 1.0) / Float64(x - 1.0)) * Float64(Float64(l * l) + Float64(2.0 * Float64(t * t)))) - Float64(l * l))))
end

MATLAB

function tmp = code(x, l, t)
	tmp = (sqrt(2.0) * t) / sqrt(((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l)));
end

Wolfram

code[x_, l_, t_] := N[(N[(N[Sqrt[2.0], $MachinePrecision] * t), $MachinePrecision] / N[Sqrt[N[(N[(N[(N[(x + 1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] + N[(2.0 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(l * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 84.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t_m \cdot \frac{\sqrt{2}}{0.5 \cdot \frac{2 \cdot \left({t_m}^{2} + {t_m}^{2}\right) + 2 \cdot {l_m}^{2}}{t_m \cdot \left(\sqrt{2} \cdot x\right)} + t_m \cdot \sqrt{2}}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 7.5 \cdot 10^{-234}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_m \leq 1.55 \cdot 10^{-186}:\\ \;\;\;\;t_m \cdot \left(\sqrt{2} \cdot \frac{1}{l_m \cdot \sqrt{\mathsf{fma}\left(2, {x}^{-2}, \frac{2}{x}\right)}}\right)\\ \mathbf{elif}\;t_m \leq 9 \cdot 10^{-160}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_m \leq 490:\\ \;\;\;\;t_m \cdot \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(2, \frac{{t_m}^{2}}{\frac{-1 + x}{x + 1}}, 2 \cdot \frac{{l_m}^{2}}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x + 1}{-1 + x}\right)}^{-0.5}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2
         (*
          t_m
          (/
           (sqrt 2.0)
           (+
            (*
             0.5
             (/
              (+ (* 2.0 (+ (pow t_m 2.0) (pow t_m 2.0))) (* 2.0 (pow l_m 2.0)))
              (* t_m (* (sqrt 2.0) x))))
            (* t_m (sqrt 2.0)))))))
   (*
    t_s
    (if (<= t_m 7.5e-234)
      t_2
      (if (<= t_m 1.55e-186)
        (*
         t_m
         (*
          (sqrt 2.0)
          (/ 1.0 (* l_m (sqrt (fma 2.0 (pow x -2.0) (/ 2.0 x)))))))
        (if (<= t_m 9e-160)
          t_2
          (if (<= t_m 490.0)
            (*
             t_m
             (/
              (sqrt 2.0)
              (sqrt
               (fma
                2.0
                (/ (pow t_m 2.0) (/ (+ -1.0 x) (+ x 1.0)))
                (* 2.0 (/ (pow l_m 2.0) x))))))
            (pow (/ (+ x 1.0) (+ -1.0 x)) -0.5))))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = t_m * (sqrt(2.0) / ((0.5 * (((2.0 * (pow(t_m, 2.0) + pow(t_m, 2.0))) + (2.0 * pow(l_m, 2.0))) / (t_m * (sqrt(2.0) * x)))) + (t_m * sqrt(2.0))));
	double tmp;
	if (t_m <= 7.5e-234) {
		tmp = t_2;
	} else if (t_m <= 1.55e-186) {
		tmp = t_m * (sqrt(2.0) * (1.0 / (l_m * sqrt(fma(2.0, pow(x, -2.0), (2.0 / x))))));
	} else if (t_m <= 9e-160) {
		tmp = t_2;
	} else if (t_m <= 490.0) {
		tmp = t_m * (sqrt(2.0) / sqrt(fma(2.0, (pow(t_m, 2.0) / ((-1.0 + x) / (x + 1.0))), (2.0 * (pow(l_m, 2.0) / x)))));
	} else {
		tmp = pow(((x + 1.0) / (-1.0 + x)), -0.5);
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(t_m * Float64(sqrt(2.0) / Float64(Float64(0.5 * Float64(Float64(Float64(2.0 * Float64((t_m ^ 2.0) + (t_m ^ 2.0))) + Float64(2.0 * (l_m ^ 2.0))) / Float64(t_m * Float64(sqrt(2.0) * x)))) + Float64(t_m * sqrt(2.0)))))
	tmp = 0.0
	if (t_m <= 7.5e-234)
		tmp = t_2;
	elseif (t_m <= 1.55e-186)
		tmp = Float64(t_m * Float64(sqrt(2.0) * Float64(1.0 / Float64(l_m * sqrt(fma(2.0, (x ^ -2.0), Float64(2.0 / x)))))));
	elseif (t_m <= 9e-160)
		tmp = t_2;
	elseif (t_m <= 490.0)
		tmp = Float64(t_m * Float64(sqrt(2.0) / sqrt(fma(2.0, Float64((t_m ^ 2.0) / Float64(Float64(-1.0 + x) / Float64(x + 1.0))), Float64(2.0 * Float64((l_m ^ 2.0) / x))))));
	else
		tmp = Float64(Float64(x + 1.0) / Float64(-1.0 + x)) ^ -0.5;
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(0.5 * N[(N[(N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] + N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 7.5e-234], t$95$2, If[LessEqual[t$95$m, 1.55e-186], N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 / N[(l$95$m * N[Sqrt[N[(2.0 * N[Power[x, -2.0], $MachinePrecision] + N[(2.0 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 9e-160], t$95$2, If[LessEqual[t$95$m, 490.0], N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(N[Power[l$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(x + 1.0), $MachinePrecision] / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]]]]]), $MachinePrecision]]

Derivation

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

Alternative 2: 85.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 6.2 \cdot 10^{-158}:\\ \;\;\;\;\frac{t_m \cdot \sqrt{2}}{l_m \cdot \sqrt{\mathsf{fma}\left(2, {x}^{-2}, \frac{2}{x}\right)}}\\ \mathbf{elif}\;t_m \leq 2800:\\ \;\;\;\;t_m \cdot \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(2, \frac{{t_m}^{2}}{\frac{-1 + x}{x + 1}}, 2 \cdot \frac{{l_m}^{2}}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x + 1}{-1 + x}\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 6.2e-158)
    (/ (* t_m (sqrt 2.0)) (* l_m (sqrt (fma 2.0 (pow x -2.0) (/ 2.0 x)))))
    (if (<= t_m 2800.0)
      (*
       t_m
       (/
        (sqrt 2.0)
        (sqrt
         (fma
          2.0
          (/ (pow t_m 2.0) (/ (+ -1.0 x) (+ x 1.0)))
          (* 2.0 (/ (pow l_m 2.0) x))))))
      (pow (/ (+ x 1.0) (+ -1.0 x)) -0.5)))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 6.2e-158) {
		tmp = (t_m * sqrt(2.0)) / (l_m * sqrt(fma(2.0, pow(x, -2.0), (2.0 / x))));
	} else if (t_m <= 2800.0) {
		tmp = t_m * (sqrt(2.0) / sqrt(fma(2.0, (pow(t_m, 2.0) / ((-1.0 + x) / (x + 1.0))), (2.0 * (pow(l_m, 2.0) / x)))));
	} else {
		tmp = pow(((x + 1.0) / (-1.0 + x)), -0.5);
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (t_m <= 6.2e-158)
		tmp = Float64(Float64(t_m * sqrt(2.0)) / Float64(l_m * sqrt(fma(2.0, (x ^ -2.0), Float64(2.0 / x)))));
	elseif (t_m <= 2800.0)
		tmp = Float64(t_m * Float64(sqrt(2.0) / sqrt(fma(2.0, Float64((t_m ^ 2.0) / Float64(Float64(-1.0 + x) / Float64(x + 1.0))), Float64(2.0 * Float64((l_m ^ 2.0) / x))))));
	else
		tmp = Float64(Float64(x + 1.0) / Float64(-1.0 + x)) ^ -0.5;
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 6.2e-158], N[(N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(l$95$m * N[Sqrt[N[(2.0 * N[Power[x, -2.0], $MachinePrecision] + N[(2.0 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2800.0], N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(N[Power[l$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(x + 1.0), $MachinePrecision] / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]]]), $MachinePrecision]

Derivation

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

Alternative 3: 79.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;l_m \cdot l_m \leq 10^{+289}:\\ \;\;\;\;{\left(\frac{x + 1}{-1 + x}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;t_m \cdot \left(\sqrt{2} \cdot \frac{1}{l_m \cdot \sqrt{\mathsf{fma}\left(2, {x}^{-2}, \frac{2}{x}\right)}}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= (* l_m l_m) 1e+289)
    (pow (/ (+ x 1.0) (+ -1.0 x)) -0.5)
    (*
     t_m
     (* (sqrt 2.0) (/ 1.0 (* l_m (sqrt (fma 2.0 (pow x -2.0) (/ 2.0 x))))))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if ((l_m * l_m) <= 1e+289) {
		tmp = pow(((x + 1.0) / (-1.0 + x)), -0.5);
	} else {
		tmp = t_m * (sqrt(2.0) * (1.0 / (l_m * sqrt(fma(2.0, pow(x, -2.0), (2.0 / x))))));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (Float64(l_m * l_m) <= 1e+289)
		tmp = Float64(Float64(x + 1.0) / Float64(-1.0 + x)) ^ -0.5;
	else
		tmp = Float64(t_m * Float64(sqrt(2.0) * Float64(1.0 / Float64(l_m * sqrt(fma(2.0, (x ^ -2.0), Float64(2.0 / x)))))));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[N[(l$95$m * l$95$m), $MachinePrecision], 1e+289], N[Power[N[(N[(x + 1.0), $MachinePrecision] / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision], N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 / N[(l$95$m * N[Sqrt[N[(2.0 * N[Power[x, -2.0], $MachinePrecision] + N[(2.0 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

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

Alternative 4: 79.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;l_m \cdot l_m \leq 10^{+289}:\\ \;\;\;\;{\left(\frac{x + 1}{-1 + x}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;t_m \cdot \frac{\sqrt{2}}{l_m \cdot \sqrt{\frac{2}{x} + \frac{2}{{x}^{2}}}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= (* l_m l_m) 1e+289)
    (pow (/ (+ x 1.0) (+ -1.0 x)) -0.5)
    (* t_m (/ (sqrt 2.0) (* l_m (sqrt (+ (/ 2.0 x) (/ 2.0 (pow x 2.0))))))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if ((l_m * l_m) <= 1e+289) {
		tmp = pow(((x + 1.0) / (-1.0 + x)), -0.5);
	} else {
		tmp = t_m * (sqrt(2.0) / (l_m * sqrt(((2.0 / x) + (2.0 / pow(x, 2.0))))));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if ((l_m * l_m) <= 1d+289) then
        tmp = ((x + 1.0d0) / ((-1.0d0) + x)) ** (-0.5d0)
    else
        tmp = t_m * (sqrt(2.0d0) / (l_m * sqrt(((2.0d0 / x) + (2.0d0 / (x ** 2.0d0))))))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if ((l_m * l_m) <= 1e+289) {
		tmp = Math.pow(((x + 1.0) / (-1.0 + x)), -0.5);
	} else {
		tmp = t_m * (Math.sqrt(2.0) / (l_m * Math.sqrt(((2.0 / x) + (2.0 / Math.pow(x, 2.0))))));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if (l_m * l_m) <= 1e+289:
		tmp = math.pow(((x + 1.0) / (-1.0 + x)), -0.5)
	else:
		tmp = t_m * (math.sqrt(2.0) / (l_m * math.sqrt(((2.0 / x) + (2.0 / math.pow(x, 2.0))))))
	return t_s * tmp

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (Float64(l_m * l_m) <= 1e+289)
		tmp = Float64(Float64(x + 1.0) / Float64(-1.0 + x)) ^ -0.5;
	else
		tmp = Float64(t_m * Float64(sqrt(2.0) / Float64(l_m * sqrt(Float64(Float64(2.0 / x) + Float64(2.0 / (x ^ 2.0)))))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if ((l_m * l_m) <= 1e+289)
		tmp = ((x + 1.0) / (-1.0 + x)) ^ -0.5;
	else
		tmp = t_m * (sqrt(2.0) / (l_m * sqrt(((2.0 / x) + (2.0 / (x ^ 2.0))))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[N[(l$95$m * l$95$m), $MachinePrecision], 1e+289], N[Power[N[(N[(x + 1.0), $MachinePrecision] / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision], N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / N[(l$95$m * N[Sqrt[N[(N[(2.0 / x), $MachinePrecision] + N[(2.0 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

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

Alternative 5: 79.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;l_m \cdot l_m \leq 10^{+289}:\\ \;\;\;\;{\left(\frac{x + 1}{-1 + x}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;t_m \cdot \frac{\sqrt{2}}{l_m \cdot \sqrt{\frac{2}{x}}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= (* l_m l_m) 1e+289)
    (pow (/ (+ x 1.0) (+ -1.0 x)) -0.5)
    (* t_m (/ (sqrt 2.0) (* l_m (sqrt (/ 2.0 x))))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if ((l_m * l_m) <= 1e+289) {
		tmp = pow(((x + 1.0) / (-1.0 + x)), -0.5);
	} else {
		tmp = t_m * (sqrt(2.0) / (l_m * sqrt((2.0 / x))));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if ((l_m * l_m) <= 1d+289) then
        tmp = ((x + 1.0d0) / ((-1.0d0) + x)) ** (-0.5d0)
    else
        tmp = t_m * (sqrt(2.0d0) / (l_m * sqrt((2.0d0 / x))))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if ((l_m * l_m) <= 1e+289) {
		tmp = Math.pow(((x + 1.0) / (-1.0 + x)), -0.5);
	} else {
		tmp = t_m * (Math.sqrt(2.0) / (l_m * Math.sqrt((2.0 / x))));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if (l_m * l_m) <= 1e+289:
		tmp = math.pow(((x + 1.0) / (-1.0 + x)), -0.5)
	else:
		tmp = t_m * (math.sqrt(2.0) / (l_m * math.sqrt((2.0 / x))))
	return t_s * tmp

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (Float64(l_m * l_m) <= 1e+289)
		tmp = Float64(Float64(x + 1.0) / Float64(-1.0 + x)) ^ -0.5;
	else
		tmp = Float64(t_m * Float64(sqrt(2.0) / Float64(l_m * sqrt(Float64(2.0 / x)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if ((l_m * l_m) <= 1e+289)
		tmp = ((x + 1.0) / (-1.0 + x)) ^ -0.5;
	else
		tmp = t_m * (sqrt(2.0) / (l_m * sqrt((2.0 / x))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[N[(l$95$m * l$95$m), $MachinePrecision], 1e+289], N[Power[N[(N[(x + 1.0), $MachinePrecision] / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision], N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / N[(l$95$m * N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

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

Alternative 6: 80.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 1 + \frac{1}{t_m}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;l_m \leq 3.1 \cdot 10^{+98}:\\ \;\;\;\;{\left(\frac{x + 1}{-1 + x}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;t_m \cdot \frac{\frac{3}{t_m}}{t_2 \cdot t_2 + \left(1 + t_2\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (+ 1.0 (/ 1.0 t_m))))
   (*
    t_s
    (if (<= l_m 3.1e+98)
      (pow (/ (+ x 1.0) (+ -1.0 x)) -0.5)
      (* t_m (/ (/ 3.0 t_m) (+ (* t_2 t_2) (+ 1.0 t_2))))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = 1.0 + (1.0 / t_m);
	double tmp;
	if (l_m <= 3.1e+98) {
		tmp = pow(((x + 1.0) / (-1.0 + x)), -0.5);
	} else {
		tmp = t_m * ((3.0 / t_m) / ((t_2 * t_2) + (1.0 + t_2)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = 1.0d0 + (1.0d0 / t_m)
    if (l_m <= 3.1d+98) then
        tmp = ((x + 1.0d0) / ((-1.0d0) + x)) ** (-0.5d0)
    else
        tmp = t_m * ((3.0d0 / t_m) / ((t_2 * t_2) + (1.0d0 + t_2)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = 1.0 + (1.0 / t_m);
	double tmp;
	if (l_m <= 3.1e+98) {
		tmp = Math.pow(((x + 1.0) / (-1.0 + x)), -0.5);
	} else {
		tmp = t_m * ((3.0 / t_m) / ((t_2 * t_2) + (1.0 + t_2)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	t_2 = 1.0 + (1.0 / t_m)
	tmp = 0
	if l_m <= 3.1e+98:
		tmp = math.pow(((x + 1.0) / (-1.0 + x)), -0.5)
	else:
		tmp = t_m * ((3.0 / t_m) / ((t_2 * t_2) + (1.0 + t_2)))
	return t_s * tmp

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(1.0 + Float64(1.0 / t_m))
	tmp = 0.0
	if (l_m <= 3.1e+98)
		tmp = Float64(Float64(x + 1.0) / Float64(-1.0 + x)) ^ -0.5;
	else
		tmp = Float64(t_m * Float64(Float64(3.0 / t_m) / Float64(Float64(t_2 * t_2) + Float64(1.0 + t_2))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	t_2 = 1.0 + (1.0 / t_m);
	tmp = 0.0;
	if (l_m <= 3.1e+98)
		tmp = ((x + 1.0) / (-1.0 + x)) ^ -0.5;
	else
		tmp = t_m * ((3.0 / t_m) / ((t_2 * t_2) + (1.0 + t_2)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(1.0 + N[(1.0 / t$95$m), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[l$95$m, 3.1e+98], N[Power[N[(N[(x + 1.0), $MachinePrecision] / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision], N[(t$95$m * N[(N[(3.0 / t$95$m), $MachinePrecision] / N[(N[(t$95$2 * t$95$2), $MachinePrecision] + N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

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

Alternative 7: 80.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 1 + \frac{1}{t_m}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;l_m \leq 2.65 \cdot 10^{+97}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;t_m \cdot \frac{\frac{3}{t_m}}{t_2 \cdot t_2 + \left(1 + t_2\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (+ 1.0 (/ 1.0 t_m))))
   (*
    t_s
    (if (<= l_m 2.65e+97)
      (sqrt (/ (+ -1.0 x) (+ x 1.0)))
      (* t_m (/ (/ 3.0 t_m) (+ (* t_2 t_2) (+ 1.0 t_2))))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = 1.0 + (1.0 / t_m);
	double tmp;
	if (l_m <= 2.65e+97) {
		tmp = sqrt(((-1.0 + x) / (x + 1.0)));
	} else {
		tmp = t_m * ((3.0 / t_m) / ((t_2 * t_2) + (1.0 + t_2)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = 1.0d0 + (1.0d0 / t_m)
    if (l_m <= 2.65d+97) then
        tmp = sqrt((((-1.0d0) + x) / (x + 1.0d0)))
    else
        tmp = t_m * ((3.0d0 / t_m) / ((t_2 * t_2) + (1.0d0 + t_2)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = 1.0 + (1.0 / t_m);
	double tmp;
	if (l_m <= 2.65e+97) {
		tmp = Math.sqrt(((-1.0 + x) / (x + 1.0)));
	} else {
		tmp = t_m * ((3.0 / t_m) / ((t_2 * t_2) + (1.0 + t_2)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	t_2 = 1.0 + (1.0 / t_m)
	tmp = 0
	if l_m <= 2.65e+97:
		tmp = math.sqrt(((-1.0 + x) / (x + 1.0)))
	else:
		tmp = t_m * ((3.0 / t_m) / ((t_2 * t_2) + (1.0 + t_2)))
	return t_s * tmp

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(1.0 + Float64(1.0 / t_m))
	tmp = 0.0
	if (l_m <= 2.65e+97)
		tmp = sqrt(Float64(Float64(-1.0 + x) / Float64(x + 1.0)));
	else
		tmp = Float64(t_m * Float64(Float64(3.0 / t_m) / Float64(Float64(t_2 * t_2) + Float64(1.0 + t_2))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	t_2 = 1.0 + (1.0 / t_m);
	tmp = 0.0;
	if (l_m <= 2.65e+97)
		tmp = sqrt(((-1.0 + x) / (x + 1.0)));
	else
		tmp = t_m * ((3.0 / t_m) / ((t_2 * t_2) + (1.0 + t_2)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(1.0 + N[(1.0 / t$95$m), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[l$95$m, 2.65e+97], N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(t$95$m * N[(N[(3.0 / t$95$m), $MachinePrecision] / N[(N[(t$95$2 * t$95$2), $MachinePrecision] + N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

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

Alternative 8: 79.4% accurate, 8.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 1 + \frac{1}{t_m}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;l_m \leq 2.95 \cdot 10^{+95}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;t_m \cdot \frac{\frac{3}{t_m}}{t_2 \cdot t_2 + \left(1 + t_2\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (+ 1.0 (/ 1.0 t_m))))
   (*
    t_s
    (if (<= l_m 2.95e+95)
      (+ 1.0 (/ -1.0 x))
      (* t_m (/ (/ 3.0 t_m) (+ (* t_2 t_2) (+ 1.0 t_2))))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = 1.0 + (1.0 / t_m);
	double tmp;
	if (l_m <= 2.95e+95) {
		tmp = 1.0 + (-1.0 / x);
	} else {
		tmp = t_m * ((3.0 / t_m) / ((t_2 * t_2) + (1.0 + t_2)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = 1.0d0 + (1.0d0 / t_m)
    if (l_m <= 2.95d+95) then
        tmp = 1.0d0 + ((-1.0d0) / x)
    else
        tmp = t_m * ((3.0d0 / t_m) / ((t_2 * t_2) + (1.0d0 + t_2)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = 1.0 + (1.0 / t_m);
	double tmp;
	if (l_m <= 2.95e+95) {
		tmp = 1.0 + (-1.0 / x);
	} else {
		tmp = t_m * ((3.0 / t_m) / ((t_2 * t_2) + (1.0 + t_2)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	t_2 = 1.0 + (1.0 / t_m)
	tmp = 0
	if l_m <= 2.95e+95:
		tmp = 1.0 + (-1.0 / x)
	else:
		tmp = t_m * ((3.0 / t_m) / ((t_2 * t_2) + (1.0 + t_2)))
	return t_s * tmp

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(1.0 + Float64(1.0 / t_m))
	tmp = 0.0
	if (l_m <= 2.95e+95)
		tmp = Float64(1.0 + Float64(-1.0 / x));
	else
		tmp = Float64(t_m * Float64(Float64(3.0 / t_m) / Float64(Float64(t_2 * t_2) + Float64(1.0 + t_2))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	t_2 = 1.0 + (1.0 / t_m);
	tmp = 0.0;
	if (l_m <= 2.95e+95)
		tmp = 1.0 + (-1.0 / x);
	else
		tmp = t_m * ((3.0 / t_m) / ((t_2 * t_2) + (1.0 + t_2)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(1.0 + N[(1.0 / t$95$m), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[l$95$m, 2.95e+95], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], N[(t$95$m * N[(N[(3.0 / t$95$m), $MachinePrecision] / N[(N[(t$95$2 * t$95$2), $MachinePrecision] + N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

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

Alternative 9: 75.6% accurate, 45.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \left(1 + \frac{-1}{x}\right) \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m) :precision binary64 (* t_s (+ 1.0 (/ -1.0 x))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	return t_s * (1.0 + (-1.0 / x));
}

Fortran

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    code = t_s * (1.0d0 + ((-1.0d0) / x))
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	return t_s * (1.0 + (-1.0 / x));
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	return t_s * (1.0 + (-1.0 / x))

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	return Float64(t_s * Float64(1.0 + Float64(-1.0 / x)))
end

MATLAB

l_m = abs(l);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, x, l_m, t_m)
	tmp = t_s * (1.0 + (-1.0 / x));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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

Alternative 10: 74.9% accurate, 225.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot 1 \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m) :precision binary64 (* t_s 1.0))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	return t_s * 1.0;
}

Fortran

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    code = t_s * 1.0d0
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	return t_s * 1.0;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	return t_s * 1.0

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	return Float64(t_s * 1.0)
end

MATLAB

l_m = abs(l);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, x, l_m, t_m)
	tmp = t_s * 1.0;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * 1.0), $MachinePrecision]

Derivation

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

Reproduce

herbie shell --seed 2024003 
(FPCore (x l t)
  :name "Toniolo and Linder, Equation (7)"
  :precision binary64
  (/ (* (sqrt 2.0) t) (sqrt (- (* (/ (+ x 1.0) (- x 1.0)) (+ (* l l) (* 2.0 (* t t)))) (* l l)))))