Average Error: 42.0 → 9.4
Time: 1.8m
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 -1.4499401141744246 \cdot 10^{+69}:\\ \;\;\;\;\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 -2.1508132600515558 \cdot 10^{-163}:\\ \;\;\;\;\sqrt{\sqrt[3]{\sqrt{2}}} \cdot \frac{\left|\sqrt[3]{\sqrt{2}}\right| \cdot \left(t \cdot \sqrt{\sqrt{2}}\right)}{\sqrt{(2 \cdot \left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) + \left(\frac{t \cdot t}{\frac{x}{4}}\right))_*}}\\ \mathbf{if}\;t \le -4.760195821999297 \cdot 10^{-295}:\\ \;\;\;\;\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 5.89249116395762 \cdot 10^{-244}:\\ \;\;\;\;\frac{\left(\left|\sqrt[3]{\sqrt{2}}\right| \cdot \left(\sqrt{\sqrt{2}} \cdot t\right)\right) \cdot \sqrt{\sqrt[3]{\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 4.668565602925388 \cdot 10^{-159}:\\ \;\;\;\;\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 4.990553787764709 \cdot 10^{+140}:\\ \;\;\;\;\sqrt{\sqrt[3]{\sqrt{2}}} \cdot \frac{\left|\sqrt[3]{\sqrt{2}}\right| \cdot \left(t \cdot \sqrt{\sqrt{2}}\right)}{\sqrt{(2 \cdot \left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) + \left(\frac{t \cdot t}{\frac{x}{4}}\right))_*}}\\ \mathbf{if}\;t \le +\infty:\\ \;\;\;\;\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{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 4 regimes

  2. if t < -1.4499401141744246e+69 or -2.1508132600515558e-163 < t < -4.760195821999297e-295

    1. Initial program 49.8

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

    2. Taylor expanded around -inf 12.0

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

    3. Applied simplify12.0

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

    if -1.4499401141744246e+69 < t < -2.1508132600515558e-163 or 4.668565602925388e-159 < t < 4.990553787764709e+140

    1. Initial program 25.2

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

    2. Taylor expanded around inf 10.9

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

    3. Applied simplify5.5

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

    5. Applied add-sqr-sqrt5.7

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

    6. Applied associate-*r*5.6

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

    8. Applied add-cube-cbrt5.6

      \[\leadsto \frac{\left(t \cdot \sqrt{\sqrt{2}}\right) \cdot \sqrt{\color{blue}{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \sqrt[3]{\sqrt{2}}}}}{\sqrt{(2 \cdot \left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) + \left(\frac{4}{x} \cdot \left(t \cdot t\right)\right))_*}}\]

    9. Applied sqrt-prod5.8

      \[\leadsto \frac{\left(t \cdot \sqrt{\sqrt{2}}\right) \cdot \color{blue}{\left(\sqrt{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}} \cdot \sqrt{\sqrt[3]{\sqrt{2}}}\right)}}{\sqrt{(2 \cdot \left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) + \left(\frac{4}{x} \cdot \left(t \cdot t\right)\right))_*}}\]

    10. Applied associate-*r*5.5

      \[\leadsto \frac{\color{blue}{\left(\left(t \cdot \sqrt{\sqrt{2}}\right) \cdot \sqrt{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}\right) \cdot \sqrt{\sqrt[3]{\sqrt{2}}}}}{\sqrt{(2 \cdot \left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) + \left(\frac{4}{x} \cdot \left(t \cdot t\right)\right))_*}}\]

    11. Applied simplify5.5

      \[\leadsto \frac{\color{blue}{\left(\left|\sqrt[3]{\sqrt{2}}\right| \cdot \left(\sqrt{\sqrt{2}} \cdot t\right)\right)} \cdot \sqrt{\sqrt[3]{\sqrt{2}}}}{\sqrt{(2 \cdot \left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) + \left(\frac{4}{x} \cdot \left(t \cdot t\right)\right))_*}}\]

    12. Taylor expanded around inf 10.9

      \[\leadsto \frac{\left(\left|\sqrt[3]{\sqrt{2}}\right| \cdot \left(\sqrt{\sqrt{2}} \cdot t\right)\right) \cdot \sqrt{\sqrt[3]{\sqrt{2}}}}{\sqrt{\color{blue}{2 \cdot {t}^{2} + \left(2 \cdot \frac{{\ell}^{2}}{x} + 4 \cdot \frac{{t}^{2}}{x}\right)}}}\]

    13. Applied simplify5.4

      \[\leadsto \color{blue}{\sqrt{\sqrt[3]{\sqrt{2}}} \cdot \frac{\left|\sqrt[3]{\sqrt{2}}\right| \cdot \left(t \cdot \sqrt{\sqrt{2}}\right)}{\sqrt{(2 \cdot \left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) + \left(\frac{t \cdot t}{\frac{x}{4}}\right))_*}}}\]

    if -4.760195821999297e-295 < t < 5.89249116395762e-244

    1. Initial program 62.1

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

    2. Taylor expanded around inf 26.1

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

    3. Applied simplify25.5

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

    5. Applied add-sqr-sqrt25.5

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

    6. Applied associate-*r*25.5

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

    8. Applied add-cube-cbrt25.5

      \[\leadsto \frac{\left(t \cdot \sqrt{\sqrt{2}}\right) \cdot \sqrt{\color{blue}{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \sqrt[3]{\sqrt{2}}}}}{\sqrt{(2 \cdot \left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) + \left(\frac{4}{x} \cdot \left(t \cdot t\right)\right))_*}}\]

    9. Applied sqrt-prod25.6

      \[\leadsto \frac{\left(t \cdot \sqrt{\sqrt{2}}\right) \cdot \color{blue}{\left(\sqrt{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}} \cdot \sqrt{\sqrt[3]{\sqrt{2}}}\right)}}{\sqrt{(2 \cdot \left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) + \left(\frac{4}{x} \cdot \left(t \cdot t\right)\right))_*}}\]

    10. Applied associate-*r*25.5

      \[\leadsto \frac{\color{blue}{\left(\left(t \cdot \sqrt{\sqrt{2}}\right) \cdot \sqrt{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}\right) \cdot \sqrt{\sqrt[3]{\sqrt{2}}}}}{\sqrt{(2 \cdot \left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) + \left(\frac{4}{x} \cdot \left(t \cdot t\right)\right))_*}}\]

    11. Applied simplify25.5

      \[\leadsto \frac{\color{blue}{\left(\left|\sqrt[3]{\sqrt{2}}\right| \cdot \left(\sqrt{\sqrt{2}} \cdot t\right)\right)} \cdot \sqrt{\sqrt[3]{\sqrt{2}}}}{\sqrt{(2 \cdot \left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) + \left(\frac{4}{x} \cdot \left(t \cdot t\right)\right))_*}}\]

    if 5.89249116395762e-244 < t < 4.668565602925388e-159 or 4.990553787764709e+140 < t

    1. Initial program 58.8

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

    2. Taylor expanded around inf 9.8

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

    3. Applied simplify9.8

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

Runtime

Time bar (total: 1.8m)Debug logProfile

herbie shell --seed '#(1070131407 1246090267 3027482374 2150728003 2026520792 2347815650)' +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)))))