Average Error: 42.9 → 8.8
Time: 3.8m
Precision: 64
Internal Precision: 1408
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -9.913317891844654 \cdot 10^{+95}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(\frac{\frac{\frac{t}{x}}{x}}{\sqrt{2}} - t \cdot \sqrt{2}\right) - \frac{\frac{2}{x}}{\sqrt{2}} \cdot \left(\frac{t}{x} + t\right)}\\ \mathbf{if}\;t \le 2.606123770091212 \cdot 10^{-233}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\left(\frac{\ell}{x} \cdot \ell + \left(t \cdot 4\right) \cdot \frac{t}{x}\right) + \left(\left(t + t\right) \cdot t + \frac{\ell}{x} \cdot \ell\right)}}\\ \mathbf{if}\;t \le 2.2532975786058163 \cdot 10^{-178}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(\frac{\frac{2}{x}}{\sqrt{2}} \cdot \left(\frac{t}{x} + t\right) + t \cdot \sqrt{2}\right) - \frac{\frac{\frac{t}{x}}{x}}{\sqrt{2}}}\\ \mathbf{if}\;t \le 5.2745772705858925 \cdot 10^{+107}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\left(\frac{\ell}{x} \cdot \ell + \left(t \cdot 4\right) \cdot \frac{t}{x}\right) + \left(\left(t + t\right) \cdot t + \frac{\ell}{x} \cdot \ell\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(\frac{\frac{2}{x}}{\sqrt{2}} \cdot \left(\frac{t}{x} + t\right) + t \cdot \sqrt{2}\right) - \frac{\frac{\frac{t}{x}}{x}}{\sqrt{2}}}\\ \end{array}\]

Error

Bits error versus x

Bits error versus l

Bits error versus t

Derivation

  1. Split input into 3 regimes

  2. if t < -9.913317891844654e+95

    1. Initial program 50.3

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

    2. Taylor expanded around -inf 3.0

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

    3. Applied simplify3.0

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

    if -9.913317891844654e+95 < t < 2.606123770091212e-233 or 2.2532975786058163e-178 < t < 5.2745772705858925e+107

    1. Initial program 35.7

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

    2. Taylor expanded around inf 16.4

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

    4. Applied unpow216.4

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

    5. Applied associate-/l*12.0

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

    7. Applied add-sqr-sqrt12.0

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

    8. Applied sqrt-prod12.2

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

    9. Applied times-frac12.2

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

    10. Taylor expanded around 0 16.5

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

    11. Applied simplify12.0

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

    if 2.606123770091212e-233 < t < 2.2532975786058163e-178 or 5.2745772705858925e+107 < t

    1. Initial program 53.8

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

    2. Taylor expanded around inf 6.2

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

    3. Applied simplify6.2

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

Runtime

Time bar (total: 3.8m)Debug logProfile

herbie shell --seed '#(1063313015 2771194459 1594909340 1344785158 2223560818 546365448)' 
(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)))))