Average Error: 43.2 → 10.1
Time: 6.8s
Precision: binary64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \leq -1.8244705587929496 \cdot 10^{+124}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\frac{t}{x \cdot x} \cdot \left(\frac{2}{2 \cdot \sqrt{2}} - \frac{2}{\sqrt{2}}\right) - \left(t \cdot \sqrt{2} + 2 \cdot \frac{t}{\sqrt{2} \cdot x}\right)}\\ \mathbf{elif}\;t \leq -6.930677520876551 \cdot 10^{-120}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{4 \cdot \left(t \cdot \frac{t}{x}\right) + 2 \cdot \left(t \cdot t + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{elif}\;t \leq -1.1631519311837101 \cdot 10^{-175}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\frac{t}{x \cdot x} \cdot \left(\frac{2}{2 \cdot \sqrt{2}} - \frac{2}{\sqrt{2}}\right) - \left(t \cdot \sqrt{2} + 2 \cdot \frac{t}{\sqrt{2} \cdot x}\right)}\\ \mathbf{elif}\;t \leq 8.649079172249573 \cdot 10^{-243} \lor \neg \left(t \leq 2.2538051772827984 \cdot 10^{-176}\right) \land t \leq 2.0450614983521334 \cdot 10^{+132}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{4 \cdot \left(t \cdot \frac{t}{x}\right) + 2 \cdot \left(t \cdot t + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2} + \left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + \frac{t}{x \cdot x} \cdot \left(\frac{2}{\sqrt{2}} - \frac{2}{2 \cdot \sqrt{2}}\right)\right)}\\ \end{array}\]
\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 -1.8244705587929496 \cdot 10^{+124}:\\
\;\;\;\;\frac{t \cdot \sqrt{2}}{\frac{t}{x \cdot x} \cdot \left(\frac{2}{2 \cdot \sqrt{2}} - \frac{2}{\sqrt{2}}\right) - \left(t \cdot \sqrt{2} + 2 \cdot \frac{t}{\sqrt{2} \cdot x}\right)}\\

\mathbf{elif}\;t \leq -6.930677520876551 \cdot 10^{-120}:\\
\;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{4 \cdot \left(t \cdot \frac{t}{x}\right) + 2 \cdot \left(t \cdot t + \ell \cdot \frac{\ell}{x}\right)}}\\

\mathbf{elif}\;t \leq -1.1631519311837101 \cdot 10^{-175}:\\
\;\;\;\;\frac{t \cdot \sqrt{2}}{\frac{t}{x \cdot x} \cdot \left(\frac{2}{2 \cdot \sqrt{2}} - \frac{2}{\sqrt{2}}\right) - \left(t \cdot \sqrt{2} + 2 \cdot \frac{t}{\sqrt{2} \cdot x}\right)}\\

\mathbf{elif}\;t \leq 8.649079172249573 \cdot 10^{-243} \lor \neg \left(t \leq 2.2538051772827984 \cdot 10^{-176}\right) \land t \leq 2.0450614983521334 \cdot 10^{+132}:\\
\;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{4 \cdot \left(t \cdot \frac{t}{x}\right) + 2 \cdot \left(t \cdot t + \ell \cdot \frac{\ell}{x}\right)}}\\

\mathbf{else}:\\
\;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2} + \left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + \frac{t}{x \cdot x} \cdot \left(\frac{2}{\sqrt{2}} - \frac{2}{2 \cdot \sqrt{2}}\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 -1.8244705587929496e+124)
   (/
    (* t (sqrt 2.0))
    (-
     (* (/ t (* x x)) (- (/ 2.0 (* 2.0 (sqrt 2.0))) (/ 2.0 (sqrt 2.0))))
     (+ (* t (sqrt 2.0)) (* 2.0 (/ t (* (sqrt 2.0) x))))))
   (if (<= t -6.930677520876551e-120)
     (/
      (* t (sqrt 2.0))
      (sqrt (+ (* 4.0 (* t (/ t x))) (* 2.0 (+ (* t t) (* l (/ l x)))))))
     (if (<= t -1.1631519311837101e-175)
       (/
        (* t (sqrt 2.0))
        (-
         (* (/ t (* x x)) (- (/ 2.0 (* 2.0 (sqrt 2.0))) (/ 2.0 (sqrt 2.0))))
         (+ (* t (sqrt 2.0)) (* 2.0 (/ t (* (sqrt 2.0) x))))))
       (if (or (<= t 8.649079172249573e-243)
               (and (not (<= t 2.2538051772827984e-176))
                    (<= t 2.0450614983521334e+132)))
         (/
          (* t (sqrt 2.0))
          (sqrt (+ (* 4.0 (* t (/ t x))) (* 2.0 (+ (* t t) (* l (/ l x)))))))
         (/
          (* t (sqrt 2.0))
          (+
           (* t (sqrt 2.0))
           (+
            (* 2.0 (/ t (* (sqrt 2.0) x)))
            (*
             (/ t (* x x))
             (- (/ 2.0 (sqrt 2.0)) (/ 2.0 (* 2.0 (sqrt 2.0)))))))))))))
double code(double x, double l, double t) {
	return (((double) (((double) sqrt(2.0)) * t)) / ((double) sqrt(((double) (((double) ((((double) (x + 1.0)) / ((double) (x - 1.0))) * ((double) (((double) (l * l)) + ((double) (2.0 * ((double) (t * t)))))))) - ((double) (l * l)))))));
}
↓
double code(double x, double l, double t) {
	double tmp;
	if ((t <= -1.8244705587929496e+124)) {
		tmp = (((double) (t * ((double) sqrt(2.0)))) / ((double) (((double) ((t / ((double) (x * x))) * ((double) ((2.0 / ((double) (2.0 * ((double) sqrt(2.0))))) - (2.0 / ((double) sqrt(2.0))))))) - ((double) (((double) (t * ((double) sqrt(2.0)))) + ((double) (2.0 * (t / ((double) (((double) sqrt(2.0)) * x))))))))));
	} else {
		double tmp_1;
		if ((t <= -6.930677520876551e-120)) {
			tmp_1 = (((double) (t * ((double) sqrt(2.0)))) / ((double) sqrt(((double) (((double) (4.0 * ((double) (t * (t / x))))) + ((double) (2.0 * ((double) (((double) (t * t)) + ((double) (l * (l / x))))))))))));
		} else {
			double tmp_2;
			if ((t <= -1.1631519311837101e-175)) {
				tmp_2 = (((double) (t * ((double) sqrt(2.0)))) / ((double) (((double) ((t / ((double) (x * x))) * ((double) ((2.0 / ((double) (2.0 * ((double) sqrt(2.0))))) - (2.0 / ((double) sqrt(2.0))))))) - ((double) (((double) (t * ((double) sqrt(2.0)))) + ((double) (2.0 * (t / ((double) (((double) sqrt(2.0)) * x))))))))));
			} else {
				double tmp_3;
				if (((t <= 8.649079172249573e-243) || (!(t <= 2.2538051772827984e-176) && (t <= 2.0450614983521334e+132)))) {
					tmp_3 = (((double) (t * ((double) sqrt(2.0)))) / ((double) sqrt(((double) (((double) (4.0 * ((double) (t * (t / x))))) + ((double) (2.0 * ((double) (((double) (t * t)) + ((double) (l * (l / x))))))))))));
				} else {
					tmp_3 = (((double) (t * ((double) sqrt(2.0)))) / ((double) (((double) (t * ((double) sqrt(2.0)))) + ((double) (((double) (2.0 * (t / ((double) (((double) sqrt(2.0)) * x))))) + ((double) ((t / ((double) (x * x))) * ((double) ((2.0 / ((double) sqrt(2.0))) - (2.0 / ((double) (2.0 * ((double) sqrt(2.0))))))))))))));
				}
				tmp_2 = tmp_3;
			}
			tmp_1 = tmp_2;
		}
		tmp = tmp_1;
	}
	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 < -1.8244705587929496e124 or -6.9306775208765514e-120 < t < -1.1631519311837101e-175

    1. Initial program Error: 53.5 bits

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

    2. Taylor expanded around -inf Error: 6.9 bits

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

    3. SimplifiedError: 6.9 bits

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

    if -1.8244705587929496e124 < t < -6.9306775208765514e-120 or -1.1631519311837101e-175 < t < 8.64907917224957334e-243 or 2.25380517728279842e-176 < t < 2.0450614983521334e132

    1. Initial program Error: 33.4 bits

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

    2. Taylor expanded around inf Error: 15.8 bits

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

    3. SimplifiedError: 11.7 bits

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

    4. Taylor expanded around 0 Error: 15.8 bits

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

    5. SimplifiedError: 11.7 bits

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

    if 8.64907917224957334e-243 < t < 2.25380517728279842e-176 or 2.0450614983521334e132 < t

    1. Initial program Error: 58.5 bits

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

    2. Taylor expanded around inf Error: 9.3 bits

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

    3. SimplifiedError: 9.3 bits

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

  4. Final simplificationError: 10.1 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.8244705587929496 \cdot 10^{+124}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\frac{t}{x \cdot x} \cdot \left(\frac{2}{2 \cdot \sqrt{2}} - \frac{2}{\sqrt{2}}\right) - \left(t \cdot \sqrt{2} + 2 \cdot \frac{t}{\sqrt{2} \cdot x}\right)}\\ \mathbf{elif}\;t \leq -6.930677520876551 \cdot 10^{-120}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{4 \cdot \left(t \cdot \frac{t}{x}\right) + 2 \cdot \left(t \cdot t + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{elif}\;t \leq -1.1631519311837101 \cdot 10^{-175}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\frac{t}{x \cdot x} \cdot \left(\frac{2}{2 \cdot \sqrt{2}} - \frac{2}{\sqrt{2}}\right) - \left(t \cdot \sqrt{2} + 2 \cdot \frac{t}{\sqrt{2} \cdot x}\right)}\\ \mathbf{elif}\;t \leq 8.649079172249573 \cdot 10^{-243} \lor \neg \left(t \leq 2.2538051772827984 \cdot 10^{-176}\right) \land t \leq 2.0450614983521334 \cdot 10^{+132}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{4 \cdot \left(t \cdot \frac{t}{x}\right) + 2 \cdot \left(t \cdot t + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2} + \left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + \frac{t}{x \cdot x} \cdot \left(\frac{2}{\sqrt{2}} - \frac{2}{2 \cdot \sqrt{2}}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020203 
(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)))))