Toniolo and Linder, Equation (7)

Average Error: 42.6 → 9.0

Time: 16.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := {\left(\frac{t}{x}\right)}^{2}\\ t_3 := \mathsf{fma}\left(4, t \cdot \frac{t}{x}, 2 \cdot \left({\left(\frac{\ell}{x}\right)}^{2} + \mathsf{fma}\left(t, t, \ell \cdot \frac{\ell}{x}\right)\right)\right)\\ \mathbf{if}\;t \leq -9.637691388536258 \cdot 10^{+147}:\\ \;\;\;\;\frac{t_1}{-t \cdot \sqrt{2 + \left(\frac{4}{x} + \frac{4}{x \cdot x}\right)}}\\ \mathbf{elif}\;t \leq 2.1563523231601638 \cdot 10^{-303}:\\ \;\;\;\;\frac{t_1}{{\left({\left(\mathsf{fma}\left(4, t_2, t_3\right)\right)}^{0.25}\right)}^{2}}\\ \mathbf{elif}\;t \leq 2.7248008057053303 \cdot 10^{-191}:\\ \;\;\;\;\frac{t_1}{t_1 + \left(0.5 \cdot \frac{\sqrt{2} \cdot {\ell}^{2}}{t \cdot x} + \frac{t_1}{x}\right)}\\ \mathbf{elif}\;t \leq 1.0565592753655027 \cdot 10^{+27}:\\ \;\;\;\;\frac{t_1}{{\left({\left(\mathsf{fma}\left(4, t_2, {\left(\sqrt{t_3}\right)}^{2}\right)\right)}^{0.25}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}\\ \end{array} \]

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)))))

↓

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (* t (sqrt 2.0)))
        (t_2 (pow (/ t x) 2.0))
        (t_3
         (fma
          4.0
          (* t (/ t x))
          (* 2.0 (+ (pow (/ l x) 2.0) (fma t t (* l (/ l x))))))))
   (if (<= t -9.637691388536258e+147)
     (/ t_1 (- (* t (sqrt (+ 2.0 (+ (/ 4.0 x) (/ 4.0 (* x x))))))))
     (if (<= t 2.1563523231601638e-303)
       (/ t_1 (pow (pow (fma 4.0 t_2 t_3) 0.25) 2.0))
       (if (<= t 2.7248008057053303e-191)
         (/
          t_1
          (+ t_1 (+ (* 0.5 (/ (* (sqrt 2.0) (pow l 2.0)) (* t x))) (/ t_1 x))))
         (if (<= t 1.0565592753655027e+27)
           (/ t_1 (pow (pow (fma 4.0 t_2 (pow (sqrt t_3) 2.0)) 0.25) 2.0))
           (/
            t_1
            (* t (sqrt (+ (/ 2.0 (- x 1.0)) (* 2.0 (/ x (- x 1.0)))))))))))))

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)));
}

↓

double code(double x, double l, double t) {
	double t_1 = t * sqrt(2.0);
	double t_2 = pow((t / x), 2.0);
	double t_3 = fma(4.0, (t * (t / x)), (2.0 * (pow((l / x), 2.0) + fma(t, t, (l * (l / x))))));
	double tmp;
	if (t <= -9.637691388536258e+147) {
		tmp = t_1 / -(t * sqrt((2.0 + ((4.0 / x) + (4.0 / (x * x))))));
	} else if (t <= 2.1563523231601638e-303) {
		tmp = t_1 / pow(pow(fma(4.0, t_2, t_3), 0.25), 2.0);
	} else if (t <= 2.7248008057053303e-191) {
		tmp = t_1 / (t_1 + ((0.5 * ((sqrt(2.0) * pow(l, 2.0)) / (t * x))) + (t_1 / x)));
	} else if (t <= 1.0565592753655027e+27) {
		tmp = t_1 / pow(pow(fma(4.0, t_2, pow(sqrt(t_3), 2.0)), 0.25), 2.0);
	} else {
		tmp = t_1 / (t * sqrt(((2.0 / (x - 1.0)) + (2.0 * (x / (x - 1.0))))));
	}
	return tmp;
}

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

↓

function code(x, l, t)
	t_1 = Float64(t * sqrt(2.0))
	t_2 = Float64(t / x) ^ 2.0
	t_3 = fma(4.0, Float64(t * Float64(t / x)), Float64(2.0 * Float64((Float64(l / x) ^ 2.0) + fma(t, t, Float64(l * Float64(l / x))))))
	tmp = 0.0
	if (t <= -9.637691388536258e+147)
		tmp = Float64(t_1 / Float64(-Float64(t * sqrt(Float64(2.0 + Float64(Float64(4.0 / x) + Float64(4.0 / Float64(x * x))))))));
	elseif (t <= 2.1563523231601638e-303)
		tmp = Float64(t_1 / ((fma(4.0, t_2, t_3) ^ 0.25) ^ 2.0));
	elseif (t <= 2.7248008057053303e-191)
		tmp = Float64(t_1 / Float64(t_1 + Float64(Float64(0.5 * Float64(Float64(sqrt(2.0) * (l ^ 2.0)) / Float64(t * x))) + Float64(t_1 / x))));
	elseif (t <= 1.0565592753655027e+27)
		tmp = Float64(t_1 / ((fma(4.0, t_2, (sqrt(t_3) ^ 2.0)) ^ 0.25) ^ 2.0));
	else
		tmp = Float64(t_1 / Float64(t * sqrt(Float64(Float64(2.0 / Float64(x - 1.0)) + Float64(2.0 * Float64(x / Float64(x - 1.0)))))));
	end
	return tmp
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]

↓

code[x_, l_, t_] := Block[{t$95$1 = N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(t / x), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(4.0 * N[(t * N[(t / x), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(N[Power[N[(l / x), $MachinePrecision], 2.0], $MachinePrecision] + N[(t * t + N[(l * N[(l / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -9.637691388536258e+147], N[(t$95$1 / (-N[(t * N[Sqrt[N[(2.0 + N[(N[(4.0 / x), $MachinePrecision] + N[(4.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[t, 2.1563523231601638e-303], N[(t$95$1 / N[Power[N[Power[N[(4.0 * t$95$2 + t$95$3), $MachinePrecision], 0.25], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.7248008057053303e-191], N[(t$95$1 / N[(t$95$1 + N[(N[(0.5 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.0565592753655027e+27], N[(t$95$1 / N[Power[N[Power[N[(4.0 * t$95$2 + N[Power[N[Sqrt[t$95$3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(t * N[Sqrt[N[(N[(2.0 / N[(x - 1.0), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(x / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Error

Bits error versus x

Bits error versus l

Bits error versus t

Derivation

1. Split input into 5 regimes

2. if t < -9.6376913885362585e147

   1. Initial program 60.2

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

   2. Simplified 60.2

      \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}} \]

   3. Taylor expanded in x around inf 61.2

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{4 \cdot \frac{{t}^{2}}{{x}^{2}} + \left(4 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot \frac{{\ell}^{2}}{{x}^{2}} + \left(2 \cdot \frac{{\ell}^{2}}{x} + 2 \cdot {t}^{2}\right)\right)\right)}}} \]

   4. Simplified 61.2

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\mathsf{fma}\left(4, \frac{t \cdot t}{x \cdot x}, \mathsf{fma}\left(4, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{x \cdot x}, 2 \cdot \left(\frac{\ell \cdot \ell}{x} + t \cdot t\right)\right)\right)\right)}}} \]

   5. Taylor expanded in t around -inf 1.9

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{-1 \cdot \left(t \cdot \sqrt{2 + \left(4 \cdot \frac{1}{x} + 4 \cdot \frac{1}{{x}^{2}}\right)}\right)}} \]

   6. Simplified 1.9

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{-t \cdot \sqrt{2 + \left(\frac{4}{x} + \frac{4}{x \cdot x}\right)}}} \]

   if -9.6376913885362585e147 < t < 2.1563523231601638e-303

   1. Initial program 35.4

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

   2. Simplified 35.4

      \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}} \]

   3. Taylor expanded in x around inf 19.1

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{4 \cdot \frac{{t}^{2}}{{x}^{2}} + \left(4 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot \frac{{\ell}^{2}}{{x}^{2}} + \left(2 \cdot \frac{{\ell}^{2}}{x} + 2 \cdot {t}^{2}\right)\right)\right)}}} \]

   4. Simplified 19.1

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\mathsf{fma}\left(4, \frac{t \cdot t}{x \cdot x}, \mathsf{fma}\left(4, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{x \cdot x}, 2 \cdot \left(\frac{\ell \cdot \ell}{x} + t \cdot t\right)\right)\right)\right)}}} \]

   5. Applied egg-rr 17.3

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{{\left({\left(\mathsf{fma}\left(4, {\left(\frac{t}{x}\right)}^{2}, \mathsf{fma}\left(4, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, {\left(\frac{\ell}{x}\right)}^{2}, 2 \cdot \mathsf{fma}\left(t, t, \frac{\ell \cdot \ell}{x}\right)\right)\right)\right)\right)}^{0.25}\right)}^{2}}} \]

   6. Applied egg-rr 12.6

      \[\leadsto \frac{\sqrt{2} \cdot t}{{\left({\left(\mathsf{fma}\left(4, {\left(\frac{t}{x}\right)}^{2}, \color{blue}{{\left(\mathsf{fma}\left(4, t \cdot \frac{t}{x}, 2 \cdot \left({\left(\frac{\ell}{x}\right)}^{2} + \mathsf{fma}\left(t, t, \ell \cdot \frac{\ell}{x}\right)\right)\right)\right)}^{1}}\right)\right)}^{0.25}\right)}^{2}} \]

   if 2.1563523231601638e-303 < t < 2.7248008057053303e-191

   1. Initial program 62.9

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

   2. Simplified 62.9

      \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}} \]

   3. Taylor expanded in x around inf 37.4

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{4 \cdot \frac{{t}^{2}}{{x}^{2}} + \left(4 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot \frac{{\ell}^{2}}{{x}^{2}} + \left(2 \cdot \frac{{\ell}^{2}}{x} + 2 \cdot {t}^{2}\right)\right)\right)}}} \]

   4. Simplified 37.4

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\mathsf{fma}\left(4, \frac{t \cdot t}{x \cdot x}, \mathsf{fma}\left(4, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{x \cdot x}, 2 \cdot \left(\frac{\ell \cdot \ell}{x} + t \cdot t\right)\right)\right)\right)}}} \]

   5. Applied egg-rr 32.1

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{{\left({\left(\mathsf{fma}\left(4, {\left(\frac{t}{x}\right)}^{2}, \mathsf{fma}\left(4, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, {\left(\frac{\ell}{x}\right)}^{2}, 2 \cdot \mathsf{fma}\left(t, t, \frac{\ell \cdot \ell}{x}\right)\right)\right)\right)\right)}^{0.25}\right)}^{2}}} \]

   6. Applied egg-rr 31.0

      \[\leadsto \frac{\sqrt{2} \cdot t}{{\color{blue}{\left({\left(\mathsf{fma}\left(4, {\left(\frac{t}{x}\right)}^{2}, \mathsf{fma}\left(4, t \cdot \frac{t}{x}, 2 \cdot \left({\left(\frac{\ell}{x}\right)}^{2} + \mathsf{fma}\left(t, t, \ell \cdot \frac{\ell}{x}\right)\right)\right)\right)\right)}^{0.125} \cdot {\left(\mathsf{fma}\left(4, {\left(\frac{t}{x}\right)}^{2}, \mathsf{fma}\left(4, t \cdot \frac{t}{x}, 2 \cdot \left({\left(\frac{\ell}{x}\right)}^{2} + \mathsf{fma}\left(t, t, \ell \cdot \frac{\ell}{x}\right)\right)\right)\right)\right)}^{0.125}\right)}}^{2}} \]

   7. Taylor expanded in x around inf 24.7

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{t \cdot \sqrt{2} + \left(0.5 \cdot \frac{{\ell}^{2} \cdot \sqrt{2}}{t \cdot x} + \frac{t \cdot \sqrt{2}}{x}\right)}} \]

   if 2.7248008057053303e-191 < t < 1.0565592753655027e27

   1. Initial program 33.7

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

   2. Simplified 33.7

      \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}} \]

   3. Taylor expanded in x around inf 13.3

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{4 \cdot \frac{{t}^{2}}{{x}^{2}} + \left(4 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot \frac{{\ell}^{2}}{{x}^{2}} + \left(2 \cdot \frac{{\ell}^{2}}{x} + 2 \cdot {t}^{2}\right)\right)\right)}}} \]

   4. Simplified 13.3

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\mathsf{fma}\left(4, \frac{t \cdot t}{x \cdot x}, \mathsf{fma}\left(4, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{x \cdot x}, 2 \cdot \left(\frac{\ell \cdot \ell}{x} + t \cdot t\right)\right)\right)\right)}}} \]

   5. Applied egg-rr 12.6

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{{\left({\left(\mathsf{fma}\left(4, {\left(\frac{t}{x}\right)}^{2}, \mathsf{fma}\left(4, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, {\left(\frac{\ell}{x}\right)}^{2}, 2 \cdot \mathsf{fma}\left(t, t, \frac{\ell \cdot \ell}{x}\right)\right)\right)\right)\right)}^{0.25}\right)}^{2}}} \]

   6. Applied egg-rr 7.7

      \[\leadsto \frac{\sqrt{2} \cdot t}{{\left({\left(\mathsf{fma}\left(4, {\left(\frac{t}{x}\right)}^{2}, \color{blue}{{\left(\sqrt{\mathsf{fma}\left(4, t \cdot \frac{t}{x}, 2 \cdot \left({\left(\frac{\ell}{x}\right)}^{2} + \mathsf{fma}\left(t, t, \ell \cdot \frac{\ell}{x}\right)\right)\right)}\right)}^{2}}\right)\right)}^{0.25}\right)}^{2}} \]

   if 1.0565592753655027e27 < t

   1. Initial program 43.2

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

   2. Simplified 43.2

      \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}} \]

   3. Taylor expanded in t around inf 4.9

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\sqrt{2 \cdot \frac{1}{x - 1} + 2 \cdot \frac{x}{x - 1}} \cdot t}} \]

   4. Simplified 4.9

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 9.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9.637691388536258 \cdot 10^{+147}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{-t \cdot \sqrt{2 + \left(\frac{4}{x} + \frac{4}{x \cdot x}\right)}}\\ \mathbf{elif}\;t \leq 2.1563523231601638 \cdot 10^{-303}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{{\left({\left(\mathsf{fma}\left(4, {\left(\frac{t}{x}\right)}^{2}, \mathsf{fma}\left(4, t \cdot \frac{t}{x}, 2 \cdot \left({\left(\frac{\ell}{x}\right)}^{2} + \mathsf{fma}\left(t, t, \ell \cdot \frac{\ell}{x}\right)\right)\right)\right)\right)}^{0.25}\right)}^{2}}\\ \mathbf{elif}\;t \leq 2.7248008057053303 \cdot 10^{-191}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2} + \left(0.5 \cdot \frac{\sqrt{2} \cdot {\ell}^{2}}{t \cdot x} + \frac{t \cdot \sqrt{2}}{x}\right)}\\ \mathbf{elif}\;t \leq 1.0565592753655027 \cdot 10^{+27}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{{\left({\left(\mathsf{fma}\left(4, {\left(\frac{t}{x}\right)}^{2}, {\left(\sqrt{\mathsf{fma}\left(4, t \cdot \frac{t}{x}, 2 \cdot \left({\left(\frac{\ell}{x}\right)}^{2} + \mathsf{fma}\left(t, t, \ell \cdot \frac{\ell}{x}\right)\right)\right)}\right)}^{2}\right)\right)}^{0.25}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022134 
(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)))))