Average Error: 43.0 → 8.8
Time: 1.1m
Precision: 64
Internal Precision: 1344
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.6756220168496518 \cdot 10^{+56}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(1 - 2\right) \cdot \frac{t}{\left(x \cdot x\right) \cdot \sqrt{2}} - \left(\sqrt{2} + \frac{\frac{2}{x}}{\sqrt{2}}\right) \cdot t}\\ \mathbf{elif}\;t \le -1.5204883632098863 \cdot 10^{-160}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right) + \left(\ell \cdot 2\right) \cdot \frac{\ell}{x}}}\\ \mathbf{elif}\;t \le -1.4622386926492231 \cdot 10^{-241}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(1 - 2\right) \cdot \frac{t}{\left(x \cdot x\right) \cdot \sqrt{2}} - \left(\sqrt{2} + \frac{\frac{2}{x}}{\sqrt{2}}\right) \cdot t}\\ \mathbf{elif}\;t \le 2.2284750911039815 \cdot 10^{-239}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\frac{\sqrt{\left(\left(\left(t \cdot t\right) \cdot x\right) \cdot \left(2 + \frac{4}{x}\right) + \left(\ell \cdot 2\right) \cdot \ell\right) \cdot \left(2 - \frac{4}{x}\right)}}{\sqrt{x \cdot \left(2 - \frac{4}{x}\right)}}}\\ \mathbf{elif}\;t \le 6.724413366958064 \cdot 10^{-173} \lor \neg \left(t \le 1.4740900332934245 \cdot 10^{+145}\right):\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\frac{\frac{t}{\sqrt{2}}}{x \cdot x} \cdot \left(2 - 1\right) + \left(\sqrt{2} + \frac{\frac{2}{x}}{\sqrt{2}}\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right) + \left(\ell \cdot 2\right) \cdot \frac{\ell}{x}}}\\ \end{array}\]

Error

Bits error versus x

Bits error versus l

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 4 regimes

  2. if t < -1.6756220168496518e+56 or -1.5204883632098863e-160 < t < -1.4622386926492231e-241

    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

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

    if -1.6756220168496518e+56 < t < -1.5204883632098863e-160 or 6.724413366958064e-173 < t < 1.4740900332934245e+145

    1. Initial program 26.9

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

    2. Taylor expanded around inf 10.8

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

    3. Simplified5.5

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

    if -1.4622386926492231e-241 < t < 2.2284750911039815e-239

    1. Initial program 61.6

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

    2. Taylor expanded around inf 31.6

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

    3. Simplified31.1

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

    5. Applied flip-+31.1

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

    6. Applied associate-*l/31.1

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

    7. Applied associate-*l/31.6

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

    8. Applied frac-add31.9

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

    9. Applied sqrt-div26.2

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

    10. Simplified26.2

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

    if 2.2284750911039815e-239 < t < 6.724413366958064e-173 or 1.4740900332934245e+145 < t

    1. Initial program 60.1

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

    2. Taylor expanded around inf 8.2

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

    3. Simplified8.2

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

  4. Final simplification8.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.6756220168496518 \cdot 10^{+56}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(1 - 2\right) \cdot \frac{t}{\left(x \cdot x\right) \cdot \sqrt{2}} - \left(\sqrt{2} + \frac{\frac{2}{x}}{\sqrt{2}}\right) \cdot t}\\ \mathbf{elif}\;t \le -1.5204883632098863 \cdot 10^{-160}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right) + \left(\ell \cdot 2\right) \cdot \frac{\ell}{x}}}\\ \mathbf{elif}\;t \le -1.4622386926492231 \cdot 10^{-241}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(1 - 2\right) \cdot \frac{t}{\left(x \cdot x\right) \cdot \sqrt{2}} - \left(\sqrt{2} + \frac{\frac{2}{x}}{\sqrt{2}}\right) \cdot t}\\ \mathbf{elif}\;t \le 2.2284750911039815 \cdot 10^{-239}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\frac{\sqrt{\left(\left(\left(t \cdot t\right) \cdot x\right) \cdot \left(2 + \frac{4}{x}\right) + \left(\ell \cdot 2\right) \cdot \ell\right) \cdot \left(2 - \frac{4}{x}\right)}}{\sqrt{x \cdot \left(2 - \frac{4}{x}\right)}}}\\ \mathbf{elif}\;t \le 6.724413366958064 \cdot 10^{-173} \lor \neg \left(t \le 1.4740900332934245 \cdot 10^{+145}\right):\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\frac{\frac{t}{\sqrt{2}}}{x \cdot x} \cdot \left(2 - 1\right) + \left(\sqrt{2} + \frac{\frac{2}{x}}{\sqrt{2}}\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right) + \left(\ell \cdot 2\right) \cdot \frac{\ell}{x}}}\\ \end{array}\]

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed 2018225 
(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)))))