Average Error: 42.5 → 9.8
Time: 2.0m
Precision: 64
Internal Precision: 1408
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.4427638579014495 \cdot 10^{+131}:\\ \;\;\;\;\frac{t}{\frac{\frac{t}{x}}{x + x} - \left(\frac{t}{1} + \frac{t}{\frac{x}{1}}\right)}\\ \mathbf{if}\;t \le 143186471723.08182:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\left(\frac{t \cdot t}{\frac{x}{4}} + \left(t \cdot t + t \cdot t\right)\right) + \frac{\ell}{x} \cdot \left(\ell + \ell\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \left(\sqrt{2} + \frac{\frac{2}{x}}{\sqrt{2}}\right) + \frac{\frac{t}{x}}{x} \cdot \left(\frac{2}{\sqrt{2}} - \frac{1}{\sqrt{2}}\right)}\\ \end{array}\]

Error

Bits error versus x

Bits error versus l

Bits error versus t

Derivation

  1. Split input into 3 regimes

  2. if t < -1.4427638579014495e+131

    1. Initial program 56.3

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

    2. Taylor expanded around inf 56.6

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

    3. Taylor expanded around -inf 63.6

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

    4. Applied simplify55.9

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

    5. Taylor expanded around -inf 2.6

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

    6. Applied simplify2.6

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

    if -1.4427638579014495e+131 < t < 143186471723.08182

    1. Initial program 38.5

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

    2. Taylor expanded around inf 18.7

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

    3. Taylor expanded around -inf 63.0

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

    4. Applied simplify14.5

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

    if 143186471723.08182 < t

    1. Initial program 41.7

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

    2. Taylor expanded around inf 5.1

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

    3. Applied simplify5.1

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

Runtime

Time bar (total: 2.0m)Debug logProfile

herbie shell --seed '#(1064300848 3212030778 2049303162 3567222883 2277747821 1384278011)' 
(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)))))