Average Error: 42.4 → 9.7
Time: 2.2m
Precision: 64
Internal Precision: 1344
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.75516826191787 \cdot 10^{+44}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\frac{\frac{t}{\sqrt{2}}}{x \cdot x} \cdot \left(1 - 2\right) - t \cdot \left(\sqrt{2} + \frac{\frac{2}{x}}{\sqrt{2}}\right)}\\ \mathbf{if}\;t \le -1.5141222160389079 \cdot 10^{-279}:\\ \;\;\;\;\frac{\left(t \cdot \sqrt{\sqrt{2}}\right) \cdot \sqrt{\sqrt{2}}}{\sqrt{\left(\frac{4}{x} + 2\right) \cdot \left(t \cdot t\right) + \frac{2 \cdot \ell}{\frac{x}{\ell}}}}\\ \mathbf{if}\;t \le 6.470054509249475 \cdot 10^{-239}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\frac{\sqrt{\left(\frac{4}{x} - 2\right) \cdot \left(\ell \cdot \frac{2}{x} + \left(2 + \frac{4}{x}\right) \cdot \frac{t}{\frac{\ell}{t}}\right)}}{\sqrt{\frac{\frac{4}{x} - 2}{\ell}}}}\\ \mathbf{if}\;t \le 3.5857671778842505 \cdot 10^{+116}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\left(\frac{4}{x} + 2\right) \cdot \left(t \cdot t\right) + \frac{\frac{2 \cdot \ell}{x}}{\frac{1}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \left(\sqrt{2} + \frac{\frac{2}{x}}{\sqrt{2}}\right) + \frac{t}{\left(x \cdot x\right) \cdot \sqrt{2}} \cdot \left(2 - 1\right)}\\ \end{array}\]

Error

Bits error versus x

Bits error versus l

Bits error versus t

Derivation

  1. Split input into 5 regimes

  2. if t < -1.75516826191787e+44

    1. Initial program 44.1

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

    2. Taylor expanded around -inf 4.7

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

    3. Applied simplify4.7

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

    if -1.75516826191787e+44 < t < -1.5141222160389079e-279

    1. Initial program 39.0

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

    2. Taylor expanded around inf 18.3

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

    3. Applied simplify13.3

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

    5. Applied add-sqr-sqrt13.4

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

    6. Applied associate-*r*13.3

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

    if -1.5141222160389079e-279 < t < 6.470054509249475e-239

    1. Initial program 62.0

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

    2. Taylor expanded around inf 28.9

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

    3. Applied simplify28.6

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

    5. Applied div-inv28.6

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

    6. Applied associate-/r*28.6

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

    8. Applied flip-+28.6

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

    9. Applied associate-*l/28.6

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

    10. Applied frac-add28.8

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

    11. Applied sqrt-div43.0

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

    12. Applied simplify36.5

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

    13. Applied simplify36.5

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

    if 6.470054509249475e-239 < t < 3.5857671778842505e+116

    1. Initial program 32.3

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

    2. Taylor expanded around inf 15.4

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

    3. Applied simplify10.4

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

    5. Applied div-inv10.4

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

    6. Applied associate-/r*10.3

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

    if 3.5857671778842505e+116 < t

    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. Taylor expanded around inf 2.3

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

    3. Applied simplify2.3

      \[\leadsto \color{blue}{\frac{t \cdot \sqrt{2}}{t \cdot \left(\sqrt{2} + \frac{\frac{2}{x}}{\sqrt{2}}\right) + \frac{t}{\left(x \cdot x\right) \cdot \sqrt{2}} \cdot \left(2 - 1\right)}}\]
  3. Recombined 5 regimes into one program.

Runtime

Time bar (total: 2.2m)Debug logProfile

herbie shell --seed '#(1071373924 2949776965 1885069702 3247780810 90874544 2263903749)' 
(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)))))