Average Error: 42.2 → 10.2
Time: 53.6s
Precision: 64
Internal Precision: 1344
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -2.086665853668608 \cdot 10^{+140}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{\frac{t}{x}}{2 \cdot x} - \frac{t}{x}\right) \cdot \frac{2}{\sqrt{2}} - \left(\sqrt{2} \cdot t + \frac{\frac{t}{x}}{x} \cdot \frac{2}{\sqrt{2}}\right)}\\ \mathbf{elif}\;t \le -1.5532231886814201 \cdot 10^{-161}:\\ \;\;\;\;\left(\sqrt[3]{\left(\sqrt{2} \cdot t\right) \cdot \frac{1}{\sqrt{\left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right) + \left(\ell \cdot 2\right) \cdot \frac{\ell}{x}}}} \cdot \sqrt[3]{\left(\sqrt{2} \cdot t\right) \cdot \frac{1}{\sqrt{\left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right) + \left(\ell \cdot 2\right) \cdot \frac{\ell}{x}}}}\right) \cdot \sqrt[3]{\left(\sqrt{2} \cdot t\right) \cdot \frac{1}{\sqrt{\left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right) + \left(\ell \cdot 2\right) \cdot \frac{\ell}{x}}}}\\ \mathbf{elif}\;t \le -3.283983308722692 \cdot 10^{-196}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{\frac{t}{x}}{2 \cdot x} - \frac{t}{x}\right) \cdot \frac{2}{\sqrt{2}} - \left(\sqrt{2} \cdot t + \frac{\frac{t}{x}}{x} \cdot \frac{2}{\sqrt{2}}\right)}\\ \mathbf{elif}\;t \le 3.197673185810754 \cdot 10^{-305}:\\ \;\;\;\;\sqrt{\sqrt{2}} \cdot \frac{\sqrt{\sqrt{2}}}{\frac{\sqrt{\left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right) + \left(\ell \cdot 2\right) \cdot \frac{\ell}{x}}}{t}}\\ \mathbf{elif}\;t \le 4.4053072651025186 \cdot 10^{-206} \lor \neg \left(t \le 1.590394549652717 \cdot 10^{-20}\right):\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{t \cdot \left(\frac{\frac{2}{x}}{\sqrt{2}} + \sqrt{2}\right) + \frac{\frac{\frac{2}{x}}{x}}{\sqrt{2}} \cdot \left(t - \frac{t}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\sqrt{\left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right) + \left(\ell \cdot 2\right) \cdot \frac{\ell}{x}}}{t}}\\ \end{array}\]

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 5 regimes

  2. if t < -2.086665853668608e+140 or -1.5532231886814201e-161 < t < -3.283983308722692e-196

    1. Initial program 59.1

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

    2. Taylor expanded around -inf 6.0

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

    3. Simplified6.0

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

    if -2.086665853668608e+140 < t < -1.5532231886814201e-161

    1. Initial program 22.4

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

    2. Taylor expanded around -inf 9.8

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

    3. Simplified4.8

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

    5. Applied associate-/l*4.8

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

    7. Applied div-inv4.8

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

    8. Applied *-un-lft-identity4.8

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

    9. Applied times-frac4.8

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

    10. Simplified4.8

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

    12. Applied add-cube-cbrt4.6

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

    if -3.283983308722692e-196 < t < 3.197673185810754e-305

    1. Initial program 61.2

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

    2. Taylor expanded around -inf 30.8

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

    3. Simplified29.3

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

    5. Applied associate-/l*29.9

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

    7. Applied *-un-lft-identity29.9

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

    8. Applied add-sqr-sqrt29.9

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

    9. Applied times-frac29.9

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

    10. Simplified29.9

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

    if 3.197673185810754e-305 < t < 4.4053072651025186e-206 or 1.590394549652717e-20 < t

    1. Initial program 44.0

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

    2. Taylor expanded around inf 11.8

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

    3. Simplified11.9

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

    if 4.4053072651025186e-206 < t < 1.590394549652717e-20

    1. Initial program 37.5

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

    2. Taylor expanded around -inf 16.6

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

    3. Simplified11.1

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

    5. Applied associate-/l*11.3

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{\ell}{x} \cdot \left(\ell \cdot 2\right) + \left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)}}{t}}}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification10.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.086665853668608 \cdot 10^{+140}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{\frac{t}{x}}{2 \cdot x} - \frac{t}{x}\right) \cdot \frac{2}{\sqrt{2}} - \left(\sqrt{2} \cdot t + \frac{\frac{t}{x}}{x} \cdot \frac{2}{\sqrt{2}}\right)}\\ \mathbf{elif}\;t \le -1.5532231886814201 \cdot 10^{-161}:\\ \;\;\;\;\left(\sqrt[3]{\left(\sqrt{2} \cdot t\right) \cdot \frac{1}{\sqrt{\left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right) + \left(\ell \cdot 2\right) \cdot \frac{\ell}{x}}}} \cdot \sqrt[3]{\left(\sqrt{2} \cdot t\right) \cdot \frac{1}{\sqrt{\left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right) + \left(\ell \cdot 2\right) \cdot \frac{\ell}{x}}}}\right) \cdot \sqrt[3]{\left(\sqrt{2} \cdot t\right) \cdot \frac{1}{\sqrt{\left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right) + \left(\ell \cdot 2\right) \cdot \frac{\ell}{x}}}}\\ \mathbf{elif}\;t \le -3.283983308722692 \cdot 10^{-196}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{\frac{t}{x}}{2 \cdot x} - \frac{t}{x}\right) \cdot \frac{2}{\sqrt{2}} - \left(\sqrt{2} \cdot t + \frac{\frac{t}{x}}{x} \cdot \frac{2}{\sqrt{2}}\right)}\\ \mathbf{elif}\;t \le 3.197673185810754 \cdot 10^{-305}:\\ \;\;\;\;\sqrt{\sqrt{2}} \cdot \frac{\sqrt{\sqrt{2}}}{\frac{\sqrt{\left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right) + \left(\ell \cdot 2\right) \cdot \frac{\ell}{x}}}{t}}\\ \mathbf{elif}\;t \le 4.4053072651025186 \cdot 10^{-206} \lor \neg \left(t \le 1.590394549652717 \cdot 10^{-20}\right):\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{t \cdot \left(\frac{\frac{2}{x}}{\sqrt{2}} + \sqrt{2}\right) + \frac{\frac{\frac{2}{x}}{x}}{\sqrt{2}} \cdot \left(t - \frac{t}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\sqrt{\left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right) + \left(\ell \cdot 2\right) \cdot \frac{\ell}{x}}}{t}}\\ \end{array}\]

Runtime

Time bar (total: 53.6s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes27.310.23.024.470.4%
herbie shell --seed 2018290 
(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)))))