Average Error: 42.5 → 10.0
Time: 51.3s
Precision: 64
Internal Precision: 1344
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -6.131586249182408 \cdot 10^{+56}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{-2 \cdot \frac{t}{x \cdot \sqrt{2}} + \sqrt{2} \cdot \left(-t\right)}\\ \mathbf{elif}\;t \le -2.4184416484155524 \cdot 10^{-181}:\\ \;\;\;\;\frac{\left(\left(\sqrt{\sqrt{2}} \cdot t\right) \cdot \sqrt{\sqrt{\sqrt{2}}}\right) \cdot \sqrt{\sqrt{\sqrt{2}}}}{\sqrt{(2 \cdot \left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) + \left(\left(t \cdot 4\right) \cdot \frac{t}{x}\right))_*}}\\ \mathbf{elif}\;t \le -2.742302503635443 \cdot 10^{-280}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{-2 \cdot \frac{t}{x \cdot \sqrt{2}} + \sqrt{2} \cdot \left(-t\right)}\\ \mathbf{elif}\;t \le 9.48909234728033 \cdot 10^{+16}:\\ \;\;\;\;\frac{\left(\left(\sqrt{\sqrt{2}} \cdot t\right) \cdot \sqrt{\sqrt{\sqrt{2}}}\right) \cdot \sqrt{\sqrt{\sqrt{2}}}}{\sqrt{(2 \cdot \left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) + \left(\left(t \cdot 4\right) \cdot \frac{t}{x}\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{(\left(\frac{\frac{2}{x}}{\sqrt{2}}\right) \cdot t + \left(\sqrt{2} \cdot t\right))_*}\\ \end{array}\]

Error

Bits error versus x

Bits error versus l

Bits error versus t

Derivation

  1. Split input into 3 regimes

  2. if t < -6.131586249182408e+56 or -2.4184416484155524e-181 < t < -2.742302503635443e-280

    1. Initial program 49.3

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

    2. Initial simplification49.3

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

    3. Taylor expanded around -inf 11.9

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

    if -6.131586249182408e+56 < t < -2.4184416484155524e-181 or -2.742302503635443e-280 < t < 9.48909234728033e+16

    1. Initial program 37.9

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

    2. Initial simplification37.9

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

    3. Taylor expanded around inf 16.5

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

    4. Simplified12.2

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

    6. Applied add-sqr-sqrt12.3

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

    7. Applied associate-*r*12.3

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

    9. Applied add-sqr-sqrt12.3

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

    10. Applied sqrt-prod12.3

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

    11. Applied associate-*r*12.2

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

    if 9.48909234728033e+16 < t

    1. Initial program 42.2

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

    2. Initial simplification42.2

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

    3. Taylor expanded around inf 4.6

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

    4. Simplified4.6

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

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -6.131586249182408 \cdot 10^{+56}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{-2 \cdot \frac{t}{x \cdot \sqrt{2}} + \sqrt{2} \cdot \left(-t\right)}\\ \mathbf{elif}\;t \le -2.4184416484155524 \cdot 10^{-181}:\\ \;\;\;\;\frac{\left(\left(\sqrt{\sqrt{2}} \cdot t\right) \cdot \sqrt{\sqrt{\sqrt{2}}}\right) \cdot \sqrt{\sqrt{\sqrt{2}}}}{\sqrt{(2 \cdot \left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) + \left(\left(t \cdot 4\right) \cdot \frac{t}{x}\right))_*}}\\ \mathbf{elif}\;t \le -2.742302503635443 \cdot 10^{-280}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{-2 \cdot \frac{t}{x \cdot \sqrt{2}} + \sqrt{2} \cdot \left(-t\right)}\\ \mathbf{elif}\;t \le 9.48909234728033 \cdot 10^{+16}:\\ \;\;\;\;\frac{\left(\left(\sqrt{\sqrt{2}} \cdot t\right) \cdot \sqrt{\sqrt{\sqrt{2}}}\right) \cdot \sqrt{\sqrt{\sqrt{2}}}}{\sqrt{(2 \cdot \left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) + \left(\left(t \cdot 4\right) \cdot \frac{t}{x}\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{(\left(\frac{\frac{2}{x}}{\sqrt{2}}\right) \cdot t + \left(\sqrt{2} \cdot t\right))_*}\\ \end{array}\]

Runtime

Time bar (total: 51.3s)Debug logProfile

herbie shell --seed 2018250 +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)))))