Average Error: 42.1 → 9.9
Time: 37.9s
Precision: 64
Internal Precision: 128
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -8.151515970407029 \cdot 10^{+107}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{\frac{t}{x}}{2 \cdot x} - \frac{t}{x}\right) \cdot \frac{2}{\sqrt{2}} - \left(\sqrt{2} \cdot t + \frac{\frac{t}{x}}{x} \cdot \frac{2}{\sqrt{2}}\right)}\\ \mathbf{elif}\;t \le -6.022916791975902 \cdot 10^{-161}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(t \cdot \sqrt[3]{\sqrt{2}}\right)}{\sqrt{\left(\left(\ell \cdot 2\right) \cdot \frac{1}{x}\right) \cdot \ell + \left(\frac{4}{x} + 2\right) \cdot \left(t \cdot t\right)}}\\ \mathbf{elif}\;t \le -4.736786190076456 \cdot 10^{-304}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{\frac{t}{x}}{2 \cdot x} - \frac{t}{x}\right) \cdot \frac{2}{\sqrt{2}} - \left(\sqrt{2} \cdot t + \frac{\frac{t}{x}}{x} \cdot \frac{2}{\sqrt{2}}\right)}\\ \mathbf{elif}\;t \le 2.096588363683501 \cdot 10^{+47}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(t \cdot \sqrt[3]{\sqrt{2}}\right)}{\sqrt{\left(\left(\ell \cdot 2\right) \cdot \frac{1}{x}\right) \cdot \ell + \left(\frac{4}{x} + 2\right) \cdot \left(t \cdot t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\frac{\frac{\frac{2}{x}}{x}}{\sqrt{2}} \cdot \left(t - \frac{t}{2}\right) + \left(\sqrt{2} + \frac{\frac{2}{x}}{\sqrt{2}}\right) \cdot t}\\ \end{array}\]

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 < -8.151515970407029e+107 or -6.022916791975902e-161 < t < -4.736786190076456e-304

    1. Initial program 54.4

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

    2. Taylor expanded around -inf 14.4

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

    3. Simplified14.4

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

    if -8.151515970407029e+107 < t < -6.022916791975902e-161 or -4.736786190076456e-304 < t < 2.096588363683501e+47

    1. Initial program 33.9

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

    2. Taylor expanded around inf 15.2

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

    3. Simplified10.6

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

    5. Applied div-inv10.6

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

    6. Applied associate-*l*10.6

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\ell \cdot \left(\frac{1}{x} \cdot \left(\ell \cdot 2\right)\right)} + \left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)}}\]
    7. Using strategy rm

    8. Applied add-cube-cbrt10.6

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

    9. Applied associate-*l*10.6

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

    if 2.096588363683501e+47 < t

    1. Initial program 43.6

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

    2. Taylor expanded around inf 3.8

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

    3. Simplified3.8

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

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -8.151515970407029 \cdot 10^{+107}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{\frac{t}{x}}{2 \cdot x} - \frac{t}{x}\right) \cdot \frac{2}{\sqrt{2}} - \left(\sqrt{2} \cdot t + \frac{\frac{t}{x}}{x} \cdot \frac{2}{\sqrt{2}}\right)}\\ \mathbf{elif}\;t \le -6.022916791975902 \cdot 10^{-161}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(t \cdot \sqrt[3]{\sqrt{2}}\right)}{\sqrt{\left(\left(\ell \cdot 2\right) \cdot \frac{1}{x}\right) \cdot \ell + \left(\frac{4}{x} + 2\right) \cdot \left(t \cdot t\right)}}\\ \mathbf{elif}\;t \le -4.736786190076456 \cdot 10^{-304}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{\frac{t}{x}}{2 \cdot x} - \frac{t}{x}\right) \cdot \frac{2}{\sqrt{2}} - \left(\sqrt{2} \cdot t + \frac{\frac{t}{x}}{x} \cdot \frac{2}{\sqrt{2}}\right)}\\ \mathbf{elif}\;t \le 2.096588363683501 \cdot 10^{+47}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(t \cdot \sqrt[3]{\sqrt{2}}\right)}{\sqrt{\left(\left(\ell \cdot 2\right) \cdot \frac{1}{x}\right) \cdot \ell + \left(\frac{4}{x} + 2\right) \cdot \left(t \cdot t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\frac{\frac{\frac{2}{x}}{x}}{\sqrt{2}} \cdot \left(t - \frac{t}{2}\right) + \left(\sqrt{2} + \frac{\frac{2}{x}}{\sqrt{2}}\right) \cdot t}\\ \end{array}\]

Reproduce

herbie shell --seed 2019021 
(FPCore (x l t)
  :name "Toniolo and Linder, Equation (7)"
  (/ (* (sqrt 2) t) (sqrt (- (* (/ (+ x 1) (- x 1)) (+ (* l l) (* 2 (* t t)))) (* l l)))))