Average Error: 42.3 → 9.6
Time: 2.3m
Precision: 64
Internal Precision: 1408
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -5.946432168711873 \cdot 10^{+111}:\\ \;\;\;\;\frac{t}{\left(-t\right) - \frac{t}{2} \cdot \frac{2}{x}}\\ \mathbf{if}\;t \le -3.6850313831549756 \cdot 10^{-172}:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \left(\left(\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot t\right) \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right)\right)}{\sqrt{(\left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) \cdot 2 + \left(\frac{4}{x} \cdot \left(t \cdot t\right)\right))_*}}\\ \mathbf{if}\;t \le -3.134779153452561 \cdot 10^{-217}:\\ \;\;\;\;\frac{t}{\left(-t\right) - \frac{t}{2} \cdot \frac{2}{x}}\\ \mathbf{if}\;t \le 1.895390259818782 \cdot 10^{-277}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{(\left(t \cdot t\right) \cdot \left(2 + \frac{4}{x}\right) + \left(\frac{\ell \cdot \ell}{\frac{x}{2}}\right))_*}}\\ \mathbf{if}\;t \le 6.648209037500556 \cdot 10^{-212}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\frac{2}{x}}{\sqrt{2}} + \sqrt{2}}\\ \mathbf{if}\;t \le 6.955611091772561 \cdot 10^{+57}:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \left(\left(\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot t\right) \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right)\right)}{\sqrt{(\left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) \cdot 2 + \left(\frac{4}{x} \cdot \left(t \cdot t\right)\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\frac{2}{x}}{\sqrt{2}} + \sqrt{2}}\\ \end{array}\]

Error

Bits error versus x

Bits error versus l

Bits error versus t

Derivation

  1. Split input into 4 regimes

  2. if t < -5.946432168711873e+111 or -3.6850313831549756e-172 < t < -3.134779153452561e-217

    1. Initial program 53.6

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

    2. Applied simplify53.6

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

    3. Taylor expanded around -inf 6.8

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

    4. Applied simplify6.8

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

    if -5.946432168711873e+111 < t < -3.6850313831549756e-172 or 6.648209037500556e-212 < t < 6.955611091772561e+57

    1. Initial program 29.9

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

    2. Applied simplify29.9

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

    3. Taylor expanded around inf 12.7

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

    4. Applied simplify7.4

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

    6. Applied add-cube-cbrt7.4

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

    7. Applied associate-*r*7.3

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

    9. Applied add-cube-cbrt7.3

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

    10. Applied associate-*r*7.4

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

    if -3.134779153452561e-217 < t < 1.895390259818782e-277

    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. Applied simplify61.4

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

    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. Applied simplify29.9

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

    6. Applied add-cube-cbrt29.9

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

    7. Applied associate-*r*29.9

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

    8. Taylor expanded around 0 29.9

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

    9. Applied simplify30.4

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

    if 1.895390259818782e-277 < t < 6.648209037500556e-212 or 6.955611091772561e+57 < t

    1. Initial program 47.5

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

    2. Applied simplify47.5

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

    3. Taylor expanded around inf 9.2

      \[\leadsto \frac{t \cdot \sqrt{2}}{\color{blue}{t \cdot \sqrt{2} + 2 \cdot \frac{t}{\sqrt{2} \cdot x}}}\]

    4. Applied simplify9.2

      \[\leadsto \color{blue}{\frac{1 \cdot \sqrt{2}}{\frac{\frac{2}{x}}{\sqrt{2}} + \sqrt{2}}}\]
  3. Recombined 4 regimes into one program.

  4. Applied simplify9.6

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;t \le -5.946432168711873 \cdot 10^{+111}:\\ \;\;\;\;\frac{t}{\left(-t\right) - \frac{t}{2} \cdot \frac{2}{x}}\\ \mathbf{if}\;t \le -3.6850313831549756 \cdot 10^{-172}:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \left(\left(\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot t\right) \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right)\right)}{\sqrt{(\left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) \cdot 2 + \left(\frac{4}{x} \cdot \left(t \cdot t\right)\right))_*}}\\ \mathbf{if}\;t \le -3.134779153452561 \cdot 10^{-217}:\\ \;\;\;\;\frac{t}{\left(-t\right) - \frac{t}{2} \cdot \frac{2}{x}}\\ \mathbf{if}\;t \le 1.895390259818782 \cdot 10^{-277}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{(\left(t \cdot t\right) \cdot \left(2 + \frac{4}{x}\right) + \left(\frac{\ell \cdot \ell}{\frac{x}{2}}\right))_*}}\\ \mathbf{if}\;t \le 6.648209037500556 \cdot 10^{-212}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\frac{2}{x}}{\sqrt{2}} + \sqrt{2}}\\ \mathbf{if}\;t \le 6.955611091772561 \cdot 10^{+57}:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \left(\left(\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot t\right) \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right)\right)}{\sqrt{(\left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) \cdot 2 + \left(\frac{4}{x} \cdot \left(t \cdot t\right)\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\frac{2}{x}}{\sqrt{2}} + \sqrt{2}}\\ \end{array}}\]

Runtime

Time bar (total: 2.3m)Debug logProfile

herbie shell --seed '#(1070833653 108281690 3330367898 3632331308 3494323072 43156186)' +o rules:numerics
(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)))))