Average Error: 43.3 → 40.6
Time: 9.4s
Precision: binary64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 6.552762192211068 \cdot 10^{+262}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{\sqrt[3]{t \cdot \sqrt{2}} \cdot \left(\sqrt[3]{t \cdot \sqrt{2}} \cdot \sqrt[3]{t \cdot \sqrt{2}}\right)}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
\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}\;\ell \cdot \ell \leq 6.552762192211068 \cdot 10^{+262}:\\
\;\;\;\;\sqrt[3]{{\left(\frac{\sqrt[3]{t \cdot \sqrt{2}} \cdot \left(\sqrt[3]{t \cdot \sqrt{2}} \cdot \sqrt[3]{t \cdot \sqrt{2}}\right)}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\right)}^{3}}\\

\mathbf{else}:\\
\;\;\;\;0\\

\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 (<= (* l l) 6.552762192211068e+262)
   (cbrt
    (pow
     (/
      (*
       (cbrt (* t (sqrt 2.0)))
       (* (cbrt (* t (sqrt 2.0))) (cbrt (* t (sqrt 2.0)))))
      (sqrt
       (- (* (/ (+ x 1.0) (- x 1.0)) (+ (* l l) (* 2.0 (* t t)))) (* l l))))
     3.0))
   0.0))
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 ((l * l) <= 6.552762192211068e+262) {
		tmp = cbrt(pow(((cbrt(t * sqrt(2.0)) * (cbrt(t * sqrt(2.0)) * cbrt(t * sqrt(2.0)))) / sqrt((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l))), 3.0));
	} else {
		tmp = 0.0;
	}
	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 2 regimes

  2. if (*.f64 l l) < 6.5527621922110679e262

    1. Initial program 37.1

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

    3. Applied add-cbrt-cube_binary64_63237.1

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

    4. Simplified37.1

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{t \cdot \sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\right)}^{3}}}\]
    5. Using strategy rm

    6. Applied add-cube-cbrt_binary64_63137.5

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

    if 6.5527621922110679e262 < (*.f64 l l)

    1. Initial program 63.5

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

    2. Taylor expanded around 0 50.8

      \[\leadsto \color{blue}{0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification40.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 6.552762192211068 \cdot 10^{+262}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{\sqrt[3]{t \cdot \sqrt{2}} \cdot \left(\sqrt[3]{t \cdot \sqrt{2}} \cdot \sqrt[3]{t \cdot \sqrt{2}}\right)}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2020270 
(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)))))