Toniolo and Linder, Equation (7)

Percentage Accurate: 33.7% → 84.2%

Time: 22.0s

Alternatives: 8

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: 33.7% 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: 84.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{{\ell}^{2}}{x}\\ t_3 := 2 + \frac{4}{x}\\ t_4 := 2 \cdot {t_m}^{2}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 7.8 \cdot 10^{-172}:\\ \;\;\;\;t_m \cdot \frac{\sqrt{2}}{\mathsf{fma}\left(0.5, \frac{2 \cdot t_2}{t_m} \cdot \sqrt{\frac{1}{t_3}}, t_m \cdot \sqrt{t_3}\right)}\\ \mathbf{elif}\;t_m \leq 6.5 \cdot 10^{+68}:\\ \;\;\;\;t_m \cdot \frac{\sqrt{2}}{\sqrt{\left(2 \cdot \frac{{t_m}^{2}}{x} + \left(t_2 + t_4\right)\right) + \frac{{\ell}^{2} + t_4}{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l t_m)
 :precision binary64
 (let* ((t_2 (/ (pow l 2.0) x))
        (t_3 (+ 2.0 (/ 4.0 x)))
        (t_4 (* 2.0 (pow t_m 2.0))))
   (*
    t_s
    (if (<= t_m 7.8e-172)
      (*
       t_m
       (/
        (sqrt 2.0)
        (fma
         0.5
         (* (/ (* 2.0 t_2) t_m) (sqrt (/ 1.0 t_3)))
         (* t_m (sqrt t_3)))))
      (if (<= t_m 6.5e+68)
        (*
         t_m
         (/
          (sqrt 2.0)
          (sqrt
           (+
            (+ (* 2.0 (/ (pow t_m 2.0) x)) (+ t_2 t_4))
            (/ (+ (pow l 2.0) t_4) x)))))
        (+ 1.0 (/ -1.0 x)))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double t_2 = pow(l, 2.0) / x;
	double t_3 = 2.0 + (4.0 / x);
	double t_4 = 2.0 * pow(t_m, 2.0);
	double tmp;
	if (t_m <= 7.8e-172) {
		tmp = t_m * (sqrt(2.0) / fma(0.5, (((2.0 * t_2) / t_m) * sqrt((1.0 / t_3))), (t_m * sqrt(t_3))));
	} else if (t_m <= 6.5e+68) {
		tmp = t_m * (sqrt(2.0) / sqrt((((2.0 * (pow(t_m, 2.0) / x)) + (t_2 + t_4)) + ((pow(l, 2.0) + t_4) / x))));
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return t_s * tmp;
}

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	t_2 = Float64((l ^ 2.0) / x)
	t_3 = Float64(2.0 + Float64(4.0 / x))
	t_4 = Float64(2.0 * (t_m ^ 2.0))
	tmp = 0.0
	if (t_m <= 7.8e-172)
		tmp = Float64(t_m * Float64(sqrt(2.0) / fma(0.5, Float64(Float64(Float64(2.0 * t_2) / t_m) * sqrt(Float64(1.0 / t_3))), Float64(t_m * sqrt(t_3)))));
	elseif (t_m <= 6.5e+68)
		tmp = Float64(t_m * Float64(sqrt(2.0) / sqrt(Float64(Float64(Float64(2.0 * Float64((t_m ^ 2.0) / x)) + Float64(t_2 + t_4)) + Float64(Float64((l ^ 2.0) + t_4) / x)))));
	else
		tmp = Float64(1.0 + Float64(-1.0 / x));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_, t$95$m_] := Block[{t$95$2 = N[(N[Power[l, 2.0], $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 + N[(4.0 / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 7.8e-172], N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / N[(0.5 * N[(N[(N[(2.0 * t$95$2), $MachinePrecision] / t$95$m), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(t$95$m * N[Sqrt[t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6.5e+68], 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 + t$95$4), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[l, 2.0], $MachinePrecision] + t$95$4), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

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

Alternative 2: 84.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{{\ell}^{2}}{x}\\ t_3 := 2 + \frac{4}{x}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 7.8 \cdot 10^{-171}:\\ \;\;\;\;t_m \cdot \frac{\sqrt{2}}{\mathsf{fma}\left(0.5, \frac{2 \cdot t_2}{t_m} \cdot \sqrt{\frac{1}{t_3}}, t_m \cdot \sqrt{t_3}\right)}\\ \mathbf{elif}\;t_m \leq 1.4 \cdot 10^{+74}:\\ \;\;\;\;t_m \cdot \frac{\sqrt{2}}{\sqrt{t_2 + \left(2 \cdot \frac{{t_m}^{2}}{x} + \left(t_2 + 2 \cdot {t_m}^{2}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l t_m)
 :precision binary64
 (let* ((t_2 (/ (pow l 2.0) x)) (t_3 (+ 2.0 (/ 4.0 x))))
   (*
    t_s
    (if (<= t_m 7.8e-171)
      (*
       t_m
       (/
        (sqrt 2.0)
        (fma
         0.5
         (* (/ (* 2.0 t_2) t_m) (sqrt (/ 1.0 t_3)))
         (* t_m (sqrt t_3)))))
      (if (<= t_m 1.4e+74)
        (*
         t_m
         (/
          (sqrt 2.0)
          (sqrt
           (+
            t_2
            (+ (* 2.0 (/ (pow t_m 2.0) x)) (+ t_2 (* 2.0 (pow t_m 2.0))))))))
        (+ 1.0 (/ -1.0 x)))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double t_2 = pow(l, 2.0) / x;
	double t_3 = 2.0 + (4.0 / x);
	double tmp;
	if (t_m <= 7.8e-171) {
		tmp = t_m * (sqrt(2.0) / fma(0.5, (((2.0 * t_2) / t_m) * sqrt((1.0 / t_3))), (t_m * sqrt(t_3))));
	} else if (t_m <= 1.4e+74) {
		tmp = t_m * (sqrt(2.0) / sqrt((t_2 + ((2.0 * (pow(t_m, 2.0) / x)) + (t_2 + (2.0 * pow(t_m, 2.0)))))));
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return t_s * tmp;
}

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	t_2 = Float64((l ^ 2.0) / x)
	t_3 = Float64(2.0 + Float64(4.0 / x))
	tmp = 0.0
	if (t_m <= 7.8e-171)
		tmp = Float64(t_m * Float64(sqrt(2.0) / fma(0.5, Float64(Float64(Float64(2.0 * t_2) / t_m) * sqrt(Float64(1.0 / t_3))), Float64(t_m * sqrt(t_3)))));
	elseif (t_m <= 1.4e+74)
		tmp = Float64(t_m * Float64(sqrt(2.0) / sqrt(Float64(t_2 + Float64(Float64(2.0 * Float64((t_m ^ 2.0) / x)) + Float64(t_2 + Float64(2.0 * (t_m ^ 2.0))))))));
	else
		tmp = Float64(1.0 + Float64(-1.0 / x));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_, t$95$m_] := Block[{t$95$2 = N[(N[Power[l, 2.0], $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 + N[(4.0 / x), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 7.8e-171], N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / N[(0.5 * N[(N[(N[(2.0 * t$95$2), $MachinePrecision] / t$95$m), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(t$95$m * N[Sqrt[t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.4e+74], N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(t$95$2 + N[(N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 + N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

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

Alternative 3: 80.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 + \frac{4}{x}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 880:\\ \;\;\;\;t_m \cdot \frac{\sqrt{2}}{\mathsf{fma}\left(0.5, \frac{2 \cdot \frac{{\ell}^{2}}{x}}{t_m} \cdot \sqrt{\frac{1}{t_2}}, t_m \cdot \sqrt{t_2}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l t_m)
 :precision binary64
 (let* ((t_2 (+ 2.0 (/ 4.0 x))))
   (*
    t_s
    (if (<= t_m 880.0)
      (*
       t_m
       (/
        (sqrt 2.0)
        (fma
         0.5
         (* (/ (* 2.0 (/ (pow l 2.0) x)) t_m) (sqrt (/ 1.0 t_2)))
         (* t_m (sqrt t_2)))))
      (+ 1.0 (/ -1.0 x))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double t_2 = 2.0 + (4.0 / x);
	double tmp;
	if (t_m <= 880.0) {
		tmp = t_m * (sqrt(2.0) / fma(0.5, (((2.0 * (pow(l, 2.0) / x)) / t_m) * sqrt((1.0 / t_2))), (t_m * sqrt(t_2))));
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return t_s * tmp;
}

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	t_2 = Float64(2.0 + Float64(4.0 / x))
	tmp = 0.0
	if (t_m <= 880.0)
		tmp = Float64(t_m * Float64(sqrt(2.0) / fma(0.5, Float64(Float64(Float64(2.0 * Float64((l ^ 2.0) / x)) / t_m) * sqrt(Float64(1.0 / t_2))), Float64(t_m * sqrt(t_2)))));
	else
		tmp = Float64(1.0 + Float64(-1.0 / x));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_, t$95$m_] := Block[{t$95$2 = N[(2.0 + N[(4.0 / x), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 880.0], N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / N[(0.5 * N[(N[(N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(t$95$m * N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

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

Alternative 4: 75.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{+277}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t_m}{\ell \cdot \sqrt{\frac{1}{x} + \frac{1}{x + -1}}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l t_m)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 1e+277)
    (+ 1.0 (/ -1.0 x))
    (* (sqrt 2.0) (/ t_m (* l (sqrt (+ (/ 1.0 x) (/ 1.0 (+ x -1.0))))))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if ((l * l) <= 1e+277) {
		tmp = 1.0 + (-1.0 / x);
	} else {
		tmp = sqrt(2.0) * (t_m / (l * sqrt(((1.0 / x) + (1.0 / (x + -1.0))))));
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if ((l * l) <= 1d+277) then
        tmp = 1.0d0 + ((-1.0d0) / x)
    else
        tmp = sqrt(2.0d0) * (t_m / (l * sqrt(((1.0d0 / x) + (1.0d0 / (x + (-1.0d0)))))))
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if ((l * l) <= 1e+277) {
		tmp = 1.0 + (-1.0 / x);
	} else {
		tmp = Math.sqrt(2.0) * (t_m / (l * Math.sqrt(((1.0 / x) + (1.0 / (x + -1.0))))));
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	tmp = 0
	if (l * l) <= 1e+277:
		tmp = 1.0 + (-1.0 / x)
	else:
		tmp = math.sqrt(2.0) * (t_m / (l * math.sqrt(((1.0 / x) + (1.0 / (x + -1.0))))))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	tmp = 0.0
	if (Float64(l * l) <= 1e+277)
		tmp = Float64(1.0 + Float64(-1.0 / x));
	else
		tmp = Float64(sqrt(2.0) * Float64(t_m / Float64(l * sqrt(Float64(Float64(1.0 / x) + Float64(1.0 / Float64(x + -1.0)))))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l, t_m)
	tmp = 0.0;
	if ((l * l) <= 1e+277)
		tmp = 1.0 + (-1.0 / x);
	else
		tmp = sqrt(2.0) * (t_m / (l * sqrt(((1.0 / x) + (1.0 / (x + -1.0))))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_, t$95$m_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 1e+277], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(l * N[Sqrt[N[(N[(1.0 / x), $MachinePrecision] + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

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

Alternative 5: 77.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 1.08 \cdot 10^{+142}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;\ell \leq 9.5 \cdot 10^{+217} \lor \neg \left(\ell \leq 5.5 \cdot 10^{+254}\right):\\ \;\;\;\;t_m \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l t_m)
 :precision binary64
 (*
  t_s
  (if (<= l 1.08e+142)
    (+ 1.0 (/ -1.0 x))
    (if (or (<= l 9.5e+217) (not (<= l 5.5e+254)))
      (* t_m (/ (sqrt x) l))
      1.0))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (l <= 1.08e+142) {
		tmp = 1.0 + (-1.0 / x);
	} else if ((l <= 9.5e+217) || !(l <= 5.5e+254)) {
		tmp = t_m * (sqrt(x) / l);
	} else {
		tmp = 1.0;
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (l <= 1.08d+142) then
        tmp = 1.0d0 + ((-1.0d0) / x)
    else if ((l <= 9.5d+217) .or. (.not. (l <= 5.5d+254))) then
        tmp = t_m * (sqrt(x) / l)
    else
        tmp = 1.0d0
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (l <= 1.08e+142) {
		tmp = 1.0 + (-1.0 / x);
	} else if ((l <= 9.5e+217) || !(l <= 5.5e+254)) {
		tmp = t_m * (Math.sqrt(x) / l);
	} else {
		tmp = 1.0;
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	tmp = 0
	if l <= 1.08e+142:
		tmp = 1.0 + (-1.0 / x)
	elif (l <= 9.5e+217) or not (l <= 5.5e+254):
		tmp = t_m * (math.sqrt(x) / l)
	else:
		tmp = 1.0
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	tmp = 0.0
	if (l <= 1.08e+142)
		tmp = Float64(1.0 + Float64(-1.0 / x));
	elseif ((l <= 9.5e+217) || !(l <= 5.5e+254))
		tmp = Float64(t_m * Float64(sqrt(x) / l));
	else
		tmp = 1.0;
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l, t_m)
	tmp = 0.0;
	if (l <= 1.08e+142)
		tmp = 1.0 + (-1.0 / x);
	elseif ((l <= 9.5e+217) || ~((l <= 5.5e+254)))
		tmp = t_m * (sqrt(x) / l);
	else
		tmp = 1.0;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_, t$95$m_] := N[(t$95$s * If[LessEqual[l, 1.08e+142], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[l, 9.5e+217], N[Not[LessEqual[l, 5.5e+254]], $MachinePrecision]], N[(t$95$m * N[(N[Sqrt[x], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 1.0]]), $MachinePrecision]

Derivation

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

Alternative 6: 76.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 5 \cdot 10^{-241}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t_m}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l t_m)
 :precision binary64
 (* t_s (if (<= t_m 5e-241) (* (sqrt x) (/ t_m l)) (+ 1.0 (/ -1.0 x)))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 5e-241) {
		tmp = sqrt(x) * (t_m / l);
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 5d-241) then
        tmp = sqrt(x) * (t_m / l)
    else
        tmp = 1.0d0 + ((-1.0d0) / x)
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 5e-241) {
		tmp = Math.sqrt(x) * (t_m / l);
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	tmp = 0
	if t_m <= 5e-241:
		tmp = math.sqrt(x) * (t_m / l)
	else:
		tmp = 1.0 + (-1.0 / x)
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	tmp = 0.0
	if (t_m <= 5e-241)
		tmp = Float64(sqrt(x) * Float64(t_m / l));
	else
		tmp = Float64(1.0 + Float64(-1.0 / x));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l, t_m)
	tmp = 0.0;
	if (t_m <= 5e-241)
		tmp = sqrt(x) * (t_m / l);
	else
		tmp = 1.0 + (-1.0 / x);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 5e-241], N[(N[Sqrt[x], $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

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

Alternative 7: 75.5% accurate, 45.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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_, 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 8: 75.0% accurate, 225.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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_, t$95$m_] := N[(t$95$s * 1.0), $MachinePrecision]

Derivation

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

Reproduce

herbie shell --seed 2023350 
(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)))))