Average Error: 42.4 → 8.9
Time: 56.0s
Precision: 64
Internal Precision: 1408
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.0262212597693044 \cdot 10^{+99}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\frac{-2}{x}}{\frac{\sqrt{2}}{1}} - \frac{\sqrt{2}}{1}}\\ \mathbf{if}\;t \le -4.4287023113340517 \cdot 10^{-159}:\\ \;\;\;\;\frac{\left(t \cdot \sqrt{\sqrt{2}}\right) \cdot \sqrt{\sqrt{2}}}{\sqrt{(\left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) \cdot 2 + \left(\frac{t \cdot 4}{\frac{x}{t}}\right))_*}}\\ \mathbf{if}\;t \le -1.464599026626934 \cdot 10^{-260}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\frac{-2}{x}}{\frac{\sqrt{2}}{1}} - \frac{\sqrt{2}}{1}}\\ \mathbf{if}\;t \le 9.394947846988656 \cdot 10^{+111}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{(\left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) \cdot 2 + \left(\frac{t \cdot 4}{\frac{x}{t}}\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{(\left(\frac{t}{x}\right) \cdot \left(\frac{2}{\sqrt{2}}\right) + \left(t \cdot \sqrt{2}\right))_*}\\ \end{array}\]

Error

Bits error versus x

Bits error versus l

Bits error versus t

Derivation

  1. Split input into 4 regimes

  2. if t < -1.0262212597693044e+99 or -4.4287023113340517e-159 < t < -1.464599026626934e-260

    1. Initial program 53.4

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

    2. Applied simplify53.4

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

    3. Taylor expanded around -inf 10.7

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

    4. Applied simplify10.6

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\frac{-2}{x}}{\frac{\sqrt{2}}{1}} - \frac{\sqrt{2}}{1}}}\]

    if -1.0262212597693044e+99 < t < -4.4287023113340517e-159

    1. Initial program 25.9

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

    2. Applied simplify25.9

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

    3. Taylor expanded around inf 9.8

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

    4. Applied simplify4.3

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

    6. Applied add-sqr-sqrt4.5

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

    7. Applied associate-*r*4.4

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

    if -1.464599026626934e-260 < t < 9.394947846988656e+111

    1. Initial program 38.3

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

    2. Applied simplify38.2

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

    3. Taylor expanded around inf 17.9

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

    4. Applied simplify13.9

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

    if 9.394947846988656e+111 < t

    1. Initial program 52.2

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

    2. Applied simplify52.2

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

    3. Taylor expanded around inf 2.8

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

    4. Applied simplify2.8

      \[\leadsto \color{blue}{\frac{t \cdot \sqrt{2}}{(\left(\frac{t}{x}\right) \cdot \left(\frac{2}{\sqrt{2}}\right) + \left(t \cdot \sqrt{2}\right))_*}}\]
  3. Recombined 4 regimes into one program.

Runtime

Time bar (total: 56.0s)Debug logProfile

herbie shell --seed '#(1064173506 2580572819 2847706409 4129882574 1125180799 1845288547)' +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)))))