Average Error: 52.7 → 19.1
Time: 38.3s
Precision: 64
Internal precision: 1408
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left({\ell}^2 + 2 \cdot {t}^2\right) - {\ell}^2}}\]
⬇
\[\begin{array}{l} \mathbf{if}\;t \le -4.305840089995044 \cdot 10^{+109}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(\frac{\frac{\frac{t}{x}}{\sqrt{2}}}{x} - t \cdot \sqrt{2}\right) - \frac{t + \frac{t}{x}}{\frac{\sqrt{2}}{\frac{2}{x}}}}\\ \mathbf{if}\;t \le 7.379194447956095 \cdot 10^{-213}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{2 \cdot \frac{\ell}{\frac{x}{\ell}} + \left(4 \cdot \frac{{t}^2}{x} + 2 \cdot {t}^2\right)}}{\sqrt{2} \cdot t}}\\ \mathbf{if}\;t \le 6.04866944637363 \cdot 10^{-159}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(\sqrt{2} \cdot t + \frac{\frac{t}{x} + t}{\frac{\sqrt{2}}{\frac{2}{x}}}\right) - \frac{\frac{t}{\sqrt{2}}}{{x}^2}}\\ \mathbf{if}\;t \le 5.680448161489771 \cdot 10^{+49}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{1}} \cdot \frac{t}{\sqrt{2 \cdot \frac{\ell}{\frac{x}{\ell}} + \left(4 \cdot \frac{{t}^2}{x} + 2 \cdot {t}^2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(\sqrt{2} \cdot t + \frac{\frac{t}{x} + t}{\frac{\sqrt{2}}{\frac{2}{x}}}\right) - \frac{\frac{t}{\sqrt{2}}}{{x}^2}}\\ \end{array}\]

Error

Bits error versus x

Bits error versus l

Bits error versus t

Derivation

  1. Split input into 4 regimes.

  2. if t < -4.305840089995044e+109

    1. Initial program 57.9

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

    2. Applied taylor 27.6

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

    3. Taylor expanded around -inf 27.6

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

    4. Applied simplify 27.6

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

    5. Applied simplify 27.6

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

    if -4.305840089995044e+109 < t < 7.379194447956095e-213

    1. Initial program 50.2

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

    2. Applied taylor 17.9

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

    3. Taylor expanded around inf 17.9

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

    5. Applied square-mult 17.9

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

    6. Applied associate-/l* 15.0

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

    8. Applied clear-num 15.2

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

    if 7.379194447956095e-213 < t < 6.04866944637363e-159 or 5.680448161489771e+49 < t

    1. Initial program 56.0

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

    2. Applied taylor 23.8

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

    3. Taylor expanded around inf 23.8

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

    4. Applied simplify 22.1

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

    5. Applied simplify 22.1

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

    if 6.04866944637363e-159 < t < 5.680448161489771e+49

    1. Initial program 42.6

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

    2. Applied taylor 10.5

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

    3. Taylor expanded around inf 10.5

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

    5. Applied square-mult 10.5

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

    6. Applied associate-/l* 7.2

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

    8. Applied *-un-lft-identity 7.2

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

    9. Applied sqrt-prod 7.2

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

    10. Applied times-frac 7.3

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{1}} \cdot \frac{t}{\sqrt{2 \cdot \frac{\ell}{\frac{x}{\ell}} + \left(4 \cdot \frac{{t}^2}{x} + 2 \cdot {t}^2\right)}}}\]
  3. Recombined 4 regimes into one program.
  4. Removed slow pow expressions

Runtime

Time bar (total: 38.3s) Debug logProfile

Please include this information when filing a bug report:

herbie shell --seed '#(1064524629 4159152179 2999149171 575749698 4006532819 692958815)'
(FPCore (x l t)
  :name "Toniolo and Linder, Equation (7)"
  (/ (* (sqrt 2) t) (sqrt (- (* (/ (+ x 1) (- x 1)) (+ (sqr l) (* 2 (sqr t)))) (sqr l)))))