Toniolo and Linder, Equation (7)

Percentage Accurate: 33.4% → 85.5%

Time: 23.4s

Alternatives: 7

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: 33.4% 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: 85.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {t_m}^{2}\\ t_3 := t_2 + {l_m}^{2}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 1.75 \cdot 10^{-225}:\\ \;\;\;\;\frac{t_m \cdot \sqrt{x}}{l_m}\\ \mathbf{elif}\;t_m \leq 1.15 \cdot 10^{-160}:\\ \;\;\;\;t_m \cdot \frac{\sqrt{2}}{0.5 \cdot \frac{t_3 + t_3}{t_m \cdot \left(x \cdot \sqrt{2}\right)} + t_m \cdot \sqrt{2}}\\ \mathbf{elif}\;t_m \leq 1.95 \cdot 10^{-31}:\\ \;\;\;\;t_m \cdot \frac{\sqrt{2}}{\sqrt{\left(2 \cdot \frac{{t_m}^{2}}{x} + \left(t_2 + \frac{{l_m}^{2}}{x}\right)\right) + \frac{t_3}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

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 (* 2.0 (pow t_m 2.0))) (t_3 (+ t_2 (pow l_m 2.0))))
   (*
    t_s
    (if (<= t_m 1.75e-225)
      (/ (* t_m (sqrt x)) l_m)
      (if (<= t_m 1.15e-160)
        (*
         t_m
         (/
          (sqrt 2.0)
          (+
           (* 0.5 (/ (+ t_3 t_3) (* t_m (* x (sqrt 2.0)))))
           (* t_m (sqrt 2.0)))))
        (if (<= t_m 1.95e-31)
          (*
           t_m
           (/
            (sqrt 2.0)
            (sqrt
             (+
              (+ (* 2.0 (/ (pow t_m 2.0) x)) (+ t_2 (/ (pow l_m 2.0) x)))
              (/ t_3 x)))))
          (sqrt (/ (+ x -1.0) (+ x 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) {
	double t_2 = 2.0 * pow(t_m, 2.0);
	double t_3 = t_2 + pow(l_m, 2.0);
	double tmp;
	if (t_m <= 1.75e-225) {
		tmp = (t_m * sqrt(x)) / l_m;
	} else if (t_m <= 1.15e-160) {
		tmp = t_m * (sqrt(2.0) / ((0.5 * ((t_3 + t_3) / (t_m * (x * sqrt(2.0))))) + (t_m * sqrt(2.0))));
	} else if (t_m <= 1.95e-31) {
		tmp = t_m * (sqrt(2.0) / sqrt((((2.0 * (pow(t_m, 2.0) / x)) + (t_2 + (pow(l_m, 2.0) / x))) + (t_3 / x))));
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	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) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_2 = 2.0d0 * (t_m ** 2.0d0)
    t_3 = t_2 + (l_m ** 2.0d0)
    if (t_m <= 1.75d-225) then
        tmp = (t_m * sqrt(x)) / l_m
    else if (t_m <= 1.15d-160) then
        tmp = t_m * (sqrt(2.0d0) / ((0.5d0 * ((t_3 + t_3) / (t_m * (x * sqrt(2.0d0))))) + (t_m * sqrt(2.0d0))))
    else if (t_m <= 1.95d-31) then
        tmp = t_m * (sqrt(2.0d0) / sqrt((((2.0d0 * ((t_m ** 2.0d0) / x)) + (t_2 + ((l_m ** 2.0d0) / x))) + (t_3 / x))))
    else
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    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 t_2 = 2.0 * Math.pow(t_m, 2.0);
	double t_3 = t_2 + Math.pow(l_m, 2.0);
	double tmp;
	if (t_m <= 1.75e-225) {
		tmp = (t_m * Math.sqrt(x)) / l_m;
	} else if (t_m <= 1.15e-160) {
		tmp = t_m * (Math.sqrt(2.0) / ((0.5 * ((t_3 + t_3) / (t_m * (x * Math.sqrt(2.0))))) + (t_m * Math.sqrt(2.0))));
	} else if (t_m <= 1.95e-31) {
		tmp = t_m * (Math.sqrt(2.0) / Math.sqrt((((2.0 * (Math.pow(t_m, 2.0) / x)) + (t_2 + (Math.pow(l_m, 2.0) / x))) + (t_3 / x))));
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	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):
	t_2 = 2.0 * math.pow(t_m, 2.0)
	t_3 = t_2 + math.pow(l_m, 2.0)
	tmp = 0
	if t_m <= 1.75e-225:
		tmp = (t_m * math.sqrt(x)) / l_m
	elif t_m <= 1.15e-160:
		tmp = t_m * (math.sqrt(2.0) / ((0.5 * ((t_3 + t_3) / (t_m * (x * math.sqrt(2.0))))) + (t_m * math.sqrt(2.0))))
	elif t_m <= 1.95e-31:
		tmp = t_m * (math.sqrt(2.0) / math.sqrt((((2.0 * (math.pow(t_m, 2.0) / x)) + (t_2 + (math.pow(l_m, 2.0) / x))) + (t_3 / x))))
	else:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	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)
	t_2 = Float64(2.0 * (t_m ^ 2.0))
	t_3 = Float64(t_2 + (l_m ^ 2.0))
	tmp = 0.0
	if (t_m <= 1.75e-225)
		tmp = Float64(Float64(t_m * sqrt(x)) / l_m);
	elseif (t_m <= 1.15e-160)
		tmp = Float64(t_m * Float64(sqrt(2.0) / Float64(Float64(0.5 * Float64(Float64(t_3 + t_3) / Float64(t_m * Float64(x * sqrt(2.0))))) + Float64(t_m * sqrt(2.0)))));
	elseif (t_m <= 1.95e-31)
		tmp = Float64(t_m * Float64(sqrt(2.0) / sqrt(Float64(Float64(Float64(2.0 * Float64((t_m ^ 2.0) / x)) + Float64(t_2 + Float64((l_m ^ 2.0) / x))) + Float64(t_3 / x)))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	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)
	t_2 = 2.0 * (t_m ^ 2.0);
	t_3 = t_2 + (l_m ^ 2.0);
	tmp = 0.0;
	if (t_m <= 1.75e-225)
		tmp = (t_m * sqrt(x)) / l_m;
	elseif (t_m <= 1.15e-160)
		tmp = t_m * (sqrt(2.0) / ((0.5 * ((t_3 + t_3) / (t_m * (x * sqrt(2.0))))) + (t_m * sqrt(2.0))));
	elseif (t_m <= 1.95e-31)
		tmp = t_m * (sqrt(2.0) / sqrt((((2.0 * ((t_m ^ 2.0) / x)) + (t_2 + ((l_m ^ 2.0) / x))) + (t_3 / x))));
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	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_] := Block[{t$95$2 = N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 + N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.75e-225], N[(N[(t$95$m * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision], If[LessEqual[t$95$m, 1.15e-160], N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(0.5 * N[(N[(t$95$3 + t$95$3), $MachinePrecision] / N[(t$95$m * N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.95e-31], N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(N[(N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 + N[(N[Power[l$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

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

Alternative 2: 84.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {t_m}^{2}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 2.45 \cdot 10^{-219}:\\ \;\;\;\;\frac{t_m \cdot \sqrt{x}}{l_m}\\ \mathbf{elif}\;t_m \leq 1.4 \cdot 10^{-161}:\\ \;\;\;\;1\\ \mathbf{elif}\;t_m \leq 7 \cdot 10^{-32}:\\ \;\;\;\;t_m \cdot \frac{\sqrt{2}}{\sqrt{\left(2 \cdot \frac{{t_m}^{2}}{x} + \left(t_2 + \frac{{l_m}^{2}}{x}\right)\right) + \frac{t_2 + {l_m}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

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 (* 2.0 (pow t_m 2.0))))
   (*
    t_s
    (if (<= t_m 2.45e-219)
      (/ (* t_m (sqrt x)) l_m)
      (if (<= t_m 1.4e-161)
        1.0
        (if (<= t_m 7e-32)
          (*
           t_m
           (/
            (sqrt 2.0)
            (sqrt
             (+
              (+ (* 2.0 (/ (pow t_m 2.0) x)) (+ t_2 (/ (pow l_m 2.0) x)))
              (/ (+ t_2 (pow l_m 2.0)) x)))))
          (sqrt (/ (+ x -1.0) (+ x 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) {
	double t_2 = 2.0 * pow(t_m, 2.0);
	double tmp;
	if (t_m <= 2.45e-219) {
		tmp = (t_m * sqrt(x)) / l_m;
	} else if (t_m <= 1.4e-161) {
		tmp = 1.0;
	} else if (t_m <= 7e-32) {
		tmp = t_m * (sqrt(2.0) / sqrt((((2.0 * (pow(t_m, 2.0) / x)) + (t_2 + (pow(l_m, 2.0) / x))) + ((t_2 + pow(l_m, 2.0)) / x))));
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	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) :: t_2
    real(8) :: tmp
    t_2 = 2.0d0 * (t_m ** 2.0d0)
    if (t_m <= 2.45d-219) then
        tmp = (t_m * sqrt(x)) / l_m
    else if (t_m <= 1.4d-161) then
        tmp = 1.0d0
    else if (t_m <= 7d-32) then
        tmp = t_m * (sqrt(2.0d0) / sqrt((((2.0d0 * ((t_m ** 2.0d0) / x)) + (t_2 + ((l_m ** 2.0d0) / x))) + ((t_2 + (l_m ** 2.0d0)) / x))))
    else
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    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 t_2 = 2.0 * Math.pow(t_m, 2.0);
	double tmp;
	if (t_m <= 2.45e-219) {
		tmp = (t_m * Math.sqrt(x)) / l_m;
	} else if (t_m <= 1.4e-161) {
		tmp = 1.0;
	} else if (t_m <= 7e-32) {
		tmp = t_m * (Math.sqrt(2.0) / Math.sqrt((((2.0 * (Math.pow(t_m, 2.0) / x)) + (t_2 + (Math.pow(l_m, 2.0) / x))) + ((t_2 + Math.pow(l_m, 2.0)) / x))));
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	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):
	t_2 = 2.0 * math.pow(t_m, 2.0)
	tmp = 0
	if t_m <= 2.45e-219:
		tmp = (t_m * math.sqrt(x)) / l_m
	elif t_m <= 1.4e-161:
		tmp = 1.0
	elif t_m <= 7e-32:
		tmp = t_m * (math.sqrt(2.0) / math.sqrt((((2.0 * (math.pow(t_m, 2.0) / x)) + (t_2 + (math.pow(l_m, 2.0) / x))) + ((t_2 + math.pow(l_m, 2.0)) / x))))
	else:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	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)
	t_2 = Float64(2.0 * (t_m ^ 2.0))
	tmp = 0.0
	if (t_m <= 2.45e-219)
		tmp = Float64(Float64(t_m * sqrt(x)) / l_m);
	elseif (t_m <= 1.4e-161)
		tmp = 1.0;
	elseif (t_m <= 7e-32)
		tmp = Float64(t_m * Float64(sqrt(2.0) / sqrt(Float64(Float64(Float64(2.0 * Float64((t_m ^ 2.0) / x)) + Float64(t_2 + Float64((l_m ^ 2.0) / x))) + Float64(Float64(t_2 + (l_m ^ 2.0)) / x)))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	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)
	t_2 = 2.0 * (t_m ^ 2.0);
	tmp = 0.0;
	if (t_m <= 2.45e-219)
		tmp = (t_m * sqrt(x)) / l_m;
	elseif (t_m <= 1.4e-161)
		tmp = 1.0;
	elseif (t_m <= 7e-32)
		tmp = t_m * (sqrt(2.0) / sqrt((((2.0 * ((t_m ^ 2.0) / x)) + (t_2 + ((l_m ^ 2.0) / x))) + ((t_2 + (l_m ^ 2.0)) / x))));
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	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_] := Block[{t$95$2 = N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.45e-219], N[(N[(t$95$m * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision], If[LessEqual[t$95$m, 1.4e-161], 1.0, If[LessEqual[t$95$m, 7e-32], N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(N[(N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 + N[(N[Power[l$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$2 + N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]]

Derivation

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

Alternative 3: 79.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t_m \cdot \sqrt{x}}{l_m}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 2.9 \cdot 10^{-223}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_m \leq 1.8 \cdot 10^{-160}:\\ \;\;\;\;1\\ \mathbf{elif}\;t_m \leq 6.2 \cdot 10^{-131}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

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 x)) l_m)))
   (*
    t_s
    (if (<= t_m 2.9e-223)
      t_2
      (if (<= t_m 1.8e-160)
        1.0
        (if (<= t_m 6.2e-131) t_2 (sqrt (/ (+ x -1.0) (+ x 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) {
	double t_2 = (t_m * sqrt(x)) / l_m;
	double tmp;
	if (t_m <= 2.9e-223) {
		tmp = t_2;
	} else if (t_m <= 1.8e-160) {
		tmp = 1.0;
	} else if (t_m <= 6.2e-131) {
		tmp = t_2;
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	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) :: t_2
    real(8) :: tmp
    t_2 = (t_m * sqrt(x)) / l_m
    if (t_m <= 2.9d-223) then
        tmp = t_2
    else if (t_m <= 1.8d-160) then
        tmp = 1.0d0
    else if (t_m <= 6.2d-131) then
        tmp = t_2
    else
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    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 t_2 = (t_m * Math.sqrt(x)) / l_m;
	double tmp;
	if (t_m <= 2.9e-223) {
		tmp = t_2;
	} else if (t_m <= 1.8e-160) {
		tmp = 1.0;
	} else if (t_m <= 6.2e-131) {
		tmp = t_2;
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	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):
	t_2 = (t_m * math.sqrt(x)) / l_m
	tmp = 0
	if t_m <= 2.9e-223:
		tmp = t_2
	elif t_m <= 1.8e-160:
		tmp = 1.0
	elif t_m <= 6.2e-131:
		tmp = t_2
	else:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	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)
	t_2 = Float64(Float64(t_m * sqrt(x)) / l_m)
	tmp = 0.0
	if (t_m <= 2.9e-223)
		tmp = t_2;
	elseif (t_m <= 1.8e-160)
		tmp = 1.0;
	elseif (t_m <= 6.2e-131)
		tmp = t_2;
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	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)
	t_2 = (t_m * sqrt(x)) / l_m;
	tmp = 0.0;
	if (t_m <= 2.9e-223)
		tmp = t_2;
	elseif (t_m <= 1.8e-160)
		tmp = 1.0;
	elseif (t_m <= 6.2e-131)
		tmp = t_2;
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	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_] := Block[{t$95$2 = N[(N[(t$95$m * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.9e-223], t$95$2, If[LessEqual[t$95$m, 1.8e-160], 1.0, If[LessEqual[t$95$m, 6.2e-131], t$95$2, N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]]

Derivation

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

Alternative 4: 79.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t_m \cdot \sqrt{x}}{l_m}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 3.5 \cdot 10^{-218}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_m \leq 2.56 \cdot 10^{-160}:\\ \;\;\;\;1\\ \mathbf{elif}\;t_m \leq 3.5 \cdot 10^{-131}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \end{array} \]

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 x)) l_m)))
   (*
    t_s
    (if (<= t_m 3.5e-218)
      t_2
      (if (<= t_m 2.56e-160)
        1.0
        (if (<= t_m 3.5e-131) t_2 (+ 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) {
	double t_2 = (t_m * sqrt(x)) / l_m;
	double tmp;
	if (t_m <= 3.5e-218) {
		tmp = t_2;
	} else if (t_m <= 2.56e-160) {
		tmp = 1.0;
	} else if (t_m <= 3.5e-131) {
		tmp = t_2;
	} else {
		tmp = 1.0 + (-1.0 / 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) :: t_2
    real(8) :: tmp
    t_2 = (t_m * sqrt(x)) / l_m
    if (t_m <= 3.5d-218) then
        tmp = t_2
    else if (t_m <= 2.56d-160) then
        tmp = 1.0d0
    else if (t_m <= 3.5d-131) then
        tmp = t_2
    else
        tmp = 1.0d0 + ((-1.0d0) / 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 t_2 = (t_m * Math.sqrt(x)) / l_m;
	double tmp;
	if (t_m <= 3.5e-218) {
		tmp = t_2;
	} else if (t_m <= 2.56e-160) {
		tmp = 1.0;
	} else if (t_m <= 3.5e-131) {
		tmp = t_2;
	} else {
		tmp = 1.0 + (-1.0 / 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):
	t_2 = (t_m * math.sqrt(x)) / l_m
	tmp = 0
	if t_m <= 3.5e-218:
		tmp = t_2
	elif t_m <= 2.56e-160:
		tmp = 1.0
	elif t_m <= 3.5e-131:
		tmp = t_2
	else:
		tmp = 1.0 + (-1.0 / 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)
	t_2 = Float64(Float64(t_m * sqrt(x)) / l_m)
	tmp = 0.0
	if (t_m <= 3.5e-218)
		tmp = t_2;
	elseif (t_m <= 2.56e-160)
		tmp = 1.0;
	elseif (t_m <= 3.5e-131)
		tmp = t_2;
	else
		tmp = Float64(1.0 + Float64(-1.0 / 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)
	t_2 = (t_m * sqrt(x)) / l_m;
	tmp = 0.0;
	if (t_m <= 3.5e-218)
		tmp = t_2;
	elseif (t_m <= 2.56e-160)
		tmp = 1.0;
	elseif (t_m <= 3.5e-131)
		tmp = t_2;
	else
		tmp = 1.0 + (-1.0 / 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_] := Block[{t$95$2 = N[(N[(t$95$m * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 3.5e-218], t$95$2, If[LessEqual[t$95$m, 2.56e-160], 1.0, If[LessEqual[t$95$m, 3.5e-131], t$95$2, N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

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

Alternative 5: 78.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 1.3 \cdot 10^{-214}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t_m}{l_m}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

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 (<= t_m 1.3e-214) (* (sqrt x) (/ t_m l_m)) (+ 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) {
	double tmp;
	if (t_m <= 1.3e-214) {
		tmp = sqrt(x) * (t_m / l_m);
	} else {
		tmp = 1.0 + (-1.0 / 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 (t_m <= 1.3d-214) then
        tmp = sqrt(x) * (t_m / l_m)
    else
        tmp = 1.0d0 + ((-1.0d0) / 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 (t_m <= 1.3e-214) {
		tmp = Math.sqrt(x) * (t_m / l_m);
	} else {
		tmp = 1.0 + (-1.0 / 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 t_m <= 1.3e-214:
		tmp = math.sqrt(x) * (t_m / l_m)
	else:
		tmp = 1.0 + (-1.0 / 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 (t_m <= 1.3e-214)
		tmp = Float64(sqrt(x) * Float64(t_m / l_m));
	else
		tmp = Float64(1.0 + Float64(-1.0 / 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 (t_m <= 1.3e-214)
		tmp = sqrt(x) * (t_m / l_m);
	else
		tmp = 1.0 + (-1.0 / 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[t$95$m, 1.3e-214], N[(N[Sqrt[x], $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

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

Alternative 6: 76.1% 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 7: 75.4% 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 2023348 
(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)))))