Average Error: 42.9 → 25.8
Time: 1.1m
Precision: 64
Internal Precision: 1344
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -2.1101592364577585 \cdot 10^{-172} \lor \neg \left(t \le 1.9129371436987896 \cdot 10^{-159}\right):\\ \;\;\;\;\frac{\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \left(\left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right) \cdot \left(t \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)\right)\right)}{\sqrt{\left(\frac{4}{x} + 2\right) \cdot \left(t \cdot t\right) + \left(\ell \cdot 2\right) \cdot \frac{\ell}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\frac{\sqrt{\left(\left(\frac{-4}{x} + 2\right) \cdot \left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)\right) \cdot \left(\left(\ell \cdot 2\right) \cdot \sqrt[3]{\ell}\right) + \left(t \cdot \left(x \cdot t\right)\right) \cdot \left(4 - \frac{4}{x} \cdot \frac{4}{x}\right)}}{\sqrt{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right) \cdot \left(2 - \frac{4}{x}\right)}}}\\ \end{array}\]

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 2 regimes

  2. if t < -2.1101592364577585e-172 or 1.9129371436987896e-159 < t

    1. Initial program 38.9

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

    2. Initial simplification38.9

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

    3. Taylor expanded around inf 30.4

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

    4. Simplified26.8

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

    6. Applied add-cube-cbrt26.8

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

    7. Applied associate-*r*26.8

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

    9. Applied add-cube-cbrt26.8

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

    10. Applied associate-*r*26.8

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

    if -2.1101592364577585e-172 < t < 1.9129371436987896e-159

    1. Initial program 61.4

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

    2. Initial simplification61.4

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

    3. Taylor expanded around inf 34.2

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

    4. Simplified32.3

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

    6. Applied add-cube-cbrt32.4

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

    7. Applied associate-*l*32.4

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

    9. Applied flip-+32.4

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

    10. Applied associate-*l/32.4

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

    11. Applied cbrt-div32.4

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

    12. Applied associate-*l/32.6

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

    13. Applied cbrt-div32.6

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

    14. Applied cbrt-div32.6

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

    15. Applied frac-times32.6

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

    16. Applied frac-times34.3

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

    17. Applied frac-add34.6

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

    18. Applied sqrt-div29.3

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

    19. Simplified21.3

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

  4. Final simplification25.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.1101592364577585 \cdot 10^{-172} \lor \neg \left(t \le 1.9129371436987896 \cdot 10^{-159}\right):\\ \;\;\;\;\frac{\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \left(\left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right) \cdot \left(t \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)\right)\right)}{\sqrt{\left(\frac{4}{x} + 2\right) \cdot \left(t \cdot t\right) + \left(\ell \cdot 2\right) \cdot \frac{\ell}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\frac{\sqrt{\left(\left(\frac{-4}{x} + 2\right) \cdot \left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)\right) \cdot \left(\left(\ell \cdot 2\right) \cdot \sqrt[3]{\ell}\right) + \left(t \cdot \left(x \cdot t\right)\right) \cdot \left(4 - \frac{4}{x} \cdot \frac{4}{x}\right)}}{\sqrt{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right) \cdot \left(2 - \frac{4}{x}\right)}}}\\ \end{array}\]

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed 2018249 
(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)))))