Average Error: 42.4 → 25.6
Time: 1.2m
Precision: 64
Internal Precision: 128
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -9.453946913464536 \cdot 10^{-185}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\left(\frac{1}{x} \cdot \left(2 \cdot \ell\right)\right) \cdot \ell + \left(t \cdot t\right) \cdot \left(\frac{4}{x} + 2\right)}}\\ \mathbf{elif}\;t \le 2.7003702308264013 \cdot 10^{-177}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\frac{\sqrt{\left(\left(2 \cdot \ell\right) \cdot \ell + \left(\frac{4}{x} + 2\right) \cdot \left(t \cdot \left(t \cdot x\right)\right)\right) \cdot \left(2 - \frac{4}{x}\right)}}{\sqrt{x \cdot \left(2 - \frac{4}{x}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot t\right) \cdot \sqrt[3]{\sqrt{2}}}{\sqrt{\left(t \cdot t\right) \cdot \left(\frac{4}{x} + 2\right) + \left(2 \cdot \ell\right) \cdot \frac{\ell}{x}}}\\ \end{array}\]

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 < -9.453946913464536e-185

    1. Initial program 39.6

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

    2. Initial simplification39.6

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

    3. Taylor expanded around inf 31.0

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

    4. Simplified27.0

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

    6. Applied div-inv27.0

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

    7. Applied associate-*l*27.0

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

    if -9.453946913464536e-185 < t < 2.7003702308264013e-177

    1. Initial program 61.1

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

    2. Initial simplification61.1

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

    3. Taylor expanded around inf 31.1

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

    4. Simplified29.5

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

    6. Applied div-inv29.5

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

    7. Applied associate-*l*29.5

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

    9. Applied flip-+29.5

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

    10. Applied associate-*l/29.5

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

    11. Applied associate-*l/29.5

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

    12. Applied associate-*r/31.1

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

    13. Applied frac-add31.5

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

    14. Applied sqrt-div27.7

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

    15. Simplified20.8

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

    if 2.7003702308264013e-177 < t

    1. Initial program 38.1

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

    2. Initial simplification38.1

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

    3. Taylor expanded around inf 29.5

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

    4. Simplified26.1

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

    6. Applied add-cube-cbrt26.1

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

    7. Applied associate-*r*26.0

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

  4. Final simplification25.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -9.453946913464536 \cdot 10^{-185}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\left(\frac{1}{x} \cdot \left(2 \cdot \ell\right)\right) \cdot \ell + \left(t \cdot t\right) \cdot \left(\frac{4}{x} + 2\right)}}\\ \mathbf{elif}\;t \le 2.7003702308264013 \cdot 10^{-177}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\frac{\sqrt{\left(\left(2 \cdot \ell\right) \cdot \ell + \left(\frac{4}{x} + 2\right) \cdot \left(t \cdot \left(t \cdot x\right)\right)\right) \cdot \left(2 - \frac{4}{x}\right)}}{\sqrt{x \cdot \left(2 - \frac{4}{x}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot t\right) \cdot \sqrt[3]{\sqrt{2}}}{\sqrt{\left(t \cdot t\right) \cdot \left(\frac{4}{x} + 2\right) + \left(2 \cdot \ell\right) \cdot \frac{\ell}{x}}}\\ \end{array}\]

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018336 
(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)))))