Toniolo and Linder, Equation (7)

Average Error: 43.3 → 8.8

Time: 9.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := \sqrt{\frac{2}{x + -1} + 2 \cdot \frac{x}{x + -1}}\\ \mathbf{if}\;t \leq -8.5 \cdot 10^{+115}:\\ \;\;\;\;\frac{t_1}{t_2 \cdot \left(-t\right)}\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{+119}:\\ \;\;\;\;\frac{t_1}{\left|\sqrt{\mathsf{fma}\left(4, {\left(\frac{t}{x}\right)}^{2}, \mathsf{fma}\left(4, \frac{t}{\frac{x}{t}}, \mathsf{fma}\left(2, {\left(\frac{\ell}{x}\right)}^{2}, 2 \cdot \mathsf{fma}\left(t, t, \frac{\ell}{\frac{x}{\ell}}\right)\right)\right)\right)}\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t \cdot t_2}\\ \end{array} \]

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 (sqrt (+ (/ 2.0 (+ x -1.0)) (* 2.0 (/ x (+ x -1.0)))))))
   (if (<= t -8.5e+115)
     (/ t_1 (* t_2 (- t)))
     (if (<= t 6.5e+119)
       (/
        t_1
        (fabs
         (sqrt
          (fma
           4.0
           (pow (/ t x) 2.0)
           (fma
            4.0
            (/ t (/ x t))
            (fma 2.0 (pow (/ l x) 2.0) (* 2.0 (fma t t (/ l (/ x l))))))))))
       (/ t_1 (* t t_2))))))

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 = sqrt(((2.0 / (x + -1.0)) + (2.0 * (x / (x + -1.0)))));
	double tmp;
	if (t <= -8.5e+115) {
		tmp = t_1 / (t_2 * -t);
	} else if (t <= 6.5e+119) {
		tmp = t_1 / fabs(sqrt(fma(4.0, pow((t / x), 2.0), fma(4.0, (t / (x / t)), fma(2.0, pow((l / x), 2.0), (2.0 * fma(t, t, (l / (x / l)))))))));
	} else {
		tmp = t_1 / (t * t_2);
	}
	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 = sqrt(Float64(Float64(2.0 / Float64(x + -1.0)) + Float64(2.0 * Float64(x / Float64(x + -1.0)))))
	tmp = 0.0
	if (t <= -8.5e+115)
		tmp = Float64(t_1 / Float64(t_2 * Float64(-t)));
	elseif (t <= 6.5e+119)
		tmp = Float64(t_1 / abs(sqrt(fma(4.0, (Float64(t / x) ^ 2.0), fma(4.0, Float64(t / Float64(x / t)), fma(2.0, (Float64(l / x) ^ 2.0), Float64(2.0 * fma(t, t, Float64(l / Float64(x / l))))))))));
	else
		tmp = Float64(t_1 / Float64(t * t_2));
	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[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]}, If[LessEqual[t, -8.5e+115], N[(t$95$1 / N[(t$95$2 * (-t)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 6.5e+119], N[(t$95$1 / N[Abs[N[Sqrt[N[(4.0 * N[Power[N[(t / x), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(t / N[(x / t), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[Power[N[(l / x), $MachinePrecision], 2.0], $MachinePrecision] + N[(2.0 * N[(t * t + N[(l / N[(x / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(t * t$95$2), $MachinePrecision]), $MachinePrecision]]]]]

Error

Bits error versus x

Bits error versus l

Bits error versus t

Derivation

1. Split input into 3 regimes

2. if t < -8.50000000000000057e115

   1. Initial program 53.3

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

   2. Simplified 53.3

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

   3. Taylor expanded in t around -inf 2.4

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

   4. Simplified 2.4

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

   if -8.50000000000000057e115 < t < 6.4999999999999997e119

   1. Initial program 37.5

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

   2. Simplified 37.5

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

   3. Taylor expanded in x around inf 19.2

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

   4. Simplified 19.2

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

   5. Applied egg-rr 12.7

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

   if 6.4999999999999997e119 < t

   1. Initial program 52.5

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

   2. Simplified 52.5

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

   3. Taylor expanded in t around inf 2.1

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

   4. Simplified 2.1

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

4. Final simplification 8.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.5 \cdot 10^{+115}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{2}{x + -1} + 2 \cdot \frac{x}{x + -1}} \cdot \left(-t\right)}\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{+119}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left|\sqrt{\mathsf{fma}\left(4, {\left(\frac{t}{x}\right)}^{2}, \mathsf{fma}\left(4, \frac{t}{\frac{x}{t}}, \mathsf{fma}\left(2, {\left(\frac{\ell}{x}\right)}^{2}, 2 \cdot \mathsf{fma}\left(t, t, \frac{\ell}{\frac{x}{\ell}}\right)\right)\right)\right)}\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{\frac{2}{x + -1} + 2 \cdot \frac{x}{x + -1}}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022162 
(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)))))