Average Error: 42.9 → 12.6
Time: 13.3s
Precision: binary64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \leq -3.5574732430533503 \cdot 10^{-258}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{-t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}\\ \mathbf{elif}\;t \leq 2.1538944959529218 \cdot 10^{+71}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t\right) + 4 \cdot \frac{t \cdot t}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{\frac{4}{x \cdot x} + \left(\frac{4}{x} + \left(2 + \frac{4}{{x}^{3}}\right)\right)}}\\ \end{array}\]
\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}
\mathbf{if}\;t \leq -3.5574732430533503 \cdot 10^{-258}:\\
\;\;\;\;\frac{t \cdot \sqrt{2}}{-t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}\\

\mathbf{elif}\;t \leq 2.1538944959529218 \cdot 10^{+71}:\\
\;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t\right) + 4 \cdot \frac{t \cdot t}{x}\right)}}\\

\mathbf{else}:\\
\;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{\frac{4}{x \cdot x} + \left(\frac{4}{x} + \left(2 + \frac{4}{{x}^{3}}\right)\right)}}\\

\end{array}
(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
 (if (<= t -3.5574732430533503e-258)
   (/
    (* t (sqrt 2.0))
    (- (* t (sqrt (+ (/ 2.0 (- x 1.0)) (* 2.0 (/ x (- x 1.0))))))))
   (if (<= t 2.1538944959529218e+71)
     (/
      (* t (sqrt 2.0))
      (sqrt
       (+ (* 2.0 (/ (* l l) x)) (+ (* 2.0 (* t t)) (* 4.0 (/ (* t t) x))))))
     (/
      (* t (sqrt 2.0))
      (*
       t
       (sqrt
        (+ (/ 4.0 (* x x)) (+ (/ 4.0 x) (+ 2.0 (/ 4.0 (pow x 3.0)))))))))))
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 tmp;
	if (t <= -3.5574732430533503e-258) {
		tmp = (t * sqrt(2.0)) / -(t * sqrt((2.0 / (x - 1.0)) + (2.0 * (x / (x - 1.0)))));
	} else if (t <= 2.1538944959529218e+71) {
		tmp = (t * sqrt(2.0)) / sqrt((2.0 * ((l * l) / x)) + ((2.0 * (t * t)) + (4.0 * ((t * t) / x))));
	} else {
		tmp = (t * sqrt(2.0)) / (t * sqrt((4.0 / (x * x)) + ((4.0 / x) + (2.0 + (4.0 / pow(x, 3.0))))));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus l

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if t < -3.5574732430533503e-258

    1. Initial program 41.1

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

    2. Taylor expanded around -inf 12.3

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

    3. Simplified12.3

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

    if -3.5574732430533503e-258 < t < 2.1538944959529218e71

    1. Initial program 42.5

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

    2. Taylor expanded around inf 19.3

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

    3. Simplified19.3

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

    if 2.1538944959529218e71 < t

    1. Initial program 47.1

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

    2. Taylor expanded around inf 46.6

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{2 \cdot \frac{{\ell}^{2}}{x} + \left(4 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot \frac{{\ell}^{2}}{{x}^{3}} + \left(2 \cdot {t}^{2} + \left(2 \cdot \frac{{\ell}^{2}}{{x}^{2}} + \left(4 \cdot \frac{{t}^{2}}{{x}^{3}} + 4 \cdot \frac{{t}^{2}}{{x}^{2}}\right)\right)\right)\right)\right)}}}\]

    3. Simplified46.6

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{2 \cdot \frac{\ell \cdot \ell}{x} + \left(4 \cdot \frac{t \cdot t}{x} + \left(2 \cdot \frac{\ell \cdot \ell}{{x}^{3}} + \left(2 \cdot \left(t \cdot t + \frac{\ell \cdot \ell}{x \cdot x}\right) + 4 \cdot \left(\frac{t \cdot t}{{x}^{3}} + \frac{t \cdot t}{x \cdot x}\right)\right)\right)\right)}}}\]

    4. Taylor expanded around inf 3.8

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\sqrt{4 \cdot \frac{1}{{x}^{2}} + \left(4 \cdot \frac{1}{x} + \left(4 \cdot \frac{1}{{x}^{3}} + 2\right)\right)} \cdot t}}\]

    5. Simplified3.8

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{t \cdot \sqrt{\frac{4}{x \cdot x} + \left(\frac{4}{x} + \left(2 + \frac{4}{{x}^{3}}\right)\right)}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification12.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.5574732430533503 \cdot 10^{-258}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{-t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}\\ \mathbf{elif}\;t \leq 2.1538944959529218 \cdot 10^{+71}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t\right) + 4 \cdot \frac{t \cdot t}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{\frac{4}{x \cdot x} + \left(\frac{4}{x} + \left(2 + \frac{4}{{x}^{3}}\right)\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021147 
(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)))))