Average Error: 42.5 → 8.5
Time: 1.5m
Precision: 64
Internal Precision: 1408
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -6.476595989115937 \cdot 10^{+128}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\frac{\frac{t}{x}}{x} \cdot \frac{1}{\sqrt{2}} - (\left(\frac{2}{\sqrt{2}}\right) \cdot \left(\frac{\frac{t}{x}}{x}\right) + \left((\left(\frac{2}{\sqrt{2}}\right) \cdot \left(\frac{t}{x}\right) + \left(t \cdot \sqrt{2}\right))_*\right))_*}\\ \mathbf{if}\;t \le -2.1572189557293083 \cdot 10^{-167}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{(2 \cdot \left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) + \left(\frac{4}{x} \cdot \left(t \cdot t\right)\right))_*}}\\ \mathbf{if}\;t \le -6.022207483182711 \cdot 10^{-202}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\frac{\frac{t}{x}}{x} \cdot \frac{1}{\sqrt{2}} - (\left(\frac{2}{\sqrt{2}}\right) \cdot \left(\frac{\frac{t}{x}}{x}\right) + \left((\left(\frac{2}{\sqrt{2}}\right) \cdot \left(\frac{t}{x}\right) + \left(t \cdot \sqrt{2}\right))_*\right))_*}\\ \mathbf{if}\;t \le 1.3124314431156888 \cdot 10^{-203}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{(2 \cdot \left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) + \left(\frac{4}{x} \cdot \left(t \cdot t\right)\right))_*}}\\ \mathbf{if}\;t \le 5.629658456474058 \cdot 10^{-166}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\frac{\frac{t}{x}}{x} \cdot \left(\frac{2}{\sqrt{2}} - \frac{1}{\sqrt{2}}\right) + (\left(\frac{t}{\sqrt{2}}\right) \cdot \left(\frac{2}{x}\right) + \left(t \cdot \sqrt{2}\right))_*}\\ \mathbf{if}\;t \le 1.7478278995590523 \cdot 10^{+64}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{(2 \cdot \left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) + \left(\frac{4}{x} \cdot \left(t \cdot t\right)\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\frac{\frac{t}{x}}{x} \cdot \left(\frac{2}{\sqrt{2}} - \frac{1}{\sqrt{2}}\right) + (\left(\frac{t}{\sqrt{2}}\right) \cdot \left(\frac{2}{x}\right) + \left(t \cdot \sqrt{2}\right))_*}\\ \end{array}\]

Error

Bits error versus x

Bits error versus l

Bits error versus t

Derivation

  1. Split input into 3 regimes

  2. if t < -6.476595989115937e+128 or -2.1572189557293083e-167 < t < -6.022207483182711e-202

    1. Initial program 56.7

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

    2. Taylor expanded around -inf 5.6

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

    3. Applied simplify5.6

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

    if -6.476595989115937e+128 < t < -2.1572189557293083e-167 or -6.022207483182711e-202 < t < 1.3124314431156888e-203 or 5.629658456474058e-166 < t < 1.7478278995590523e+64

    1. Initial program 34.8

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

    2. Taylor expanded around inf 15.5

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

    3. Applied simplify11.0

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

    if 1.3124314431156888e-203 < t < 5.629658456474058e-166 or 1.7478278995590523e+64 < t

    1. Initial program 47.9

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

    2. Taylor expanded around inf 5.5

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

    3. Applied simplify5.5

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

Runtime

Time bar (total: 1.5m)Debug logProfile

herbie shell --seed '#(1064397287 3527694221 3797617954 1138343853 2854031332 1153838279)' +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)))))