Average Error: 43.0 → 9.2
Time: 58.7s
Precision: 64
Internal Precision: 1344
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.6756220168496518 \cdot 10^{+56}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{(\left(\frac{t}{x \cdot x}\right) \cdot \left(\frac{1}{\sqrt{2}} - \frac{2}{\sqrt{2}}\right) + \left((\left(\frac{2}{x}\right) \cdot \left(\frac{-t}{\sqrt{2}}\right) + \left(\left(-t\right) \cdot \sqrt{2}\right))_*\right))_*}\\ \mathbf{elif}\;t \le -1.5204883632098863 \cdot 10^{-160}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{(2 \cdot \left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) + \left(\left(4 \cdot t\right) \cdot \frac{t}{x}\right))_*}}\\ \mathbf{elif}\;t \le -3.52127919595648 \cdot 10^{-242}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{(\left(\frac{t}{x \cdot x}\right) \cdot \left(\frac{1}{\sqrt{2}} - \frac{2}{\sqrt{2}}\right) + \left((\left(\frac{2}{x}\right) \cdot \left(\frac{-t}{\sqrt{2}}\right) + \left(\left(-t\right) \cdot \sqrt{2}\right))_*\right))_*}\\ \mathbf{elif}\;t \le 3.4147974846556525 \cdot 10^{-248}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\sqrt{(2 \cdot \left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) + \left(\left(4 \cdot t\right) \cdot \frac{t}{x}\right))_*}} \cdot \sqrt{\sqrt{(2 \cdot \left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) + \left(\left(4 \cdot t\right) \cdot \frac{t}{x}\right))_*}}}\\ \mathbf{elif}\;t \le 1.9558584224129961 \cdot 10^{-168} \lor \neg \left(t \le 1.1777566477755354 \cdot 10^{+146}\right):\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{(\left(\frac{2}{\sqrt{2}}\right) \cdot \left(\frac{t}{x \cdot x} + \frac{t}{x}\right) + \left((\left(\frac{-1}{\sqrt{2}}\right) \cdot \left(\frac{t}{x \cdot x}\right) + \left(\sqrt{2} \cdot t\right))_*\right))_*}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{(2 \cdot \left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) + \left(\left(4 \cdot t\right) \cdot \frac{t}{x}\right))_*}}\\ \end{array}\]

Error

Bits error versus x

Bits error versus l

Bits error versus t

Derivation

  1. Split input into 4 regimes

  2. if t < -1.6756220168496518e+56 or -1.5204883632098863e-160 < t < -3.52127919595648e-242

    1. Initial program 48.8

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

    2. Taylor expanded around -inf 9.0

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

    3. Simplified9.0

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

    if -1.6756220168496518e+56 < t < -1.5204883632098863e-160 or 1.9558584224129961e-168 < t < 1.1777566477755354e+146

    1. Initial program 26.7

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

    2. Taylor expanded around inf 10.6

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

    3. Simplified5.4

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

    if -3.52127919595648e-242 < t < 3.4147974846556525e-248

    1. Initial program 61.7

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

    2. Taylor expanded around inf 31.3

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

    3. Simplified31.0

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

    5. Applied add-sqr-sqrt31.0

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

    6. Applied sqrt-prod31.1

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

    if 3.4147974846556525e-248 < t < 1.9558584224129961e-168 or 1.1777566477755354e+146 < t

    1. Initial program 60.3

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

    2. Taylor expanded around inf 9.4

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

    3. Simplified9.4

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{(\left(\frac{2}{\sqrt{2}}\right) \cdot \left(\frac{t}{x} + \frac{t}{x \cdot x}\right) + \left((\left(\frac{-1}{\sqrt{2}}\right) \cdot \left(\frac{t}{x \cdot x}\right) + \left(\sqrt{2} \cdot t\right))_*\right))_*}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification9.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.6756220168496518 \cdot 10^{+56}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{(\left(\frac{t}{x \cdot x}\right) \cdot \left(\frac{1}{\sqrt{2}} - \frac{2}{\sqrt{2}}\right) + \left((\left(\frac{2}{x}\right) \cdot \left(\frac{-t}{\sqrt{2}}\right) + \left(\left(-t\right) \cdot \sqrt{2}\right))_*\right))_*}\\ \mathbf{elif}\;t \le -1.5204883632098863 \cdot 10^{-160}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{(2 \cdot \left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) + \left(\left(4 \cdot t\right) \cdot \frac{t}{x}\right))_*}}\\ \mathbf{elif}\;t \le -3.52127919595648 \cdot 10^{-242}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{(\left(\frac{t}{x \cdot x}\right) \cdot \left(\frac{1}{\sqrt{2}} - \frac{2}{\sqrt{2}}\right) + \left((\left(\frac{2}{x}\right) \cdot \left(\frac{-t}{\sqrt{2}}\right) + \left(\left(-t\right) \cdot \sqrt{2}\right))_*\right))_*}\\ \mathbf{elif}\;t \le 3.4147974846556525 \cdot 10^{-248}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\sqrt{(2 \cdot \left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) + \left(\left(4 \cdot t\right) \cdot \frac{t}{x}\right))_*}} \cdot \sqrt{\sqrt{(2 \cdot \left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) + \left(\left(4 \cdot t\right) \cdot \frac{t}{x}\right))_*}}}\\ \mathbf{elif}\;t \le 1.9558584224129961 \cdot 10^{-168} \lor \neg \left(t \le 1.1777566477755354 \cdot 10^{+146}\right):\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{(\left(\frac{2}{\sqrt{2}}\right) \cdot \left(\frac{t}{x \cdot x} + \frac{t}{x}\right) + \left((\left(\frac{-1}{\sqrt{2}}\right) \cdot \left(\frac{t}{x \cdot x}\right) + \left(\sqrt{2} \cdot t\right))_*\right))_*}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{(2 \cdot \left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) + \left(\left(4 \cdot t\right) \cdot \frac{t}{x}\right))_*}}\\ \end{array}\]

Runtime

Time bar (total: 58.7s)Debug logProfile

herbie shell --seed 2018225 +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)))))