Average Error: 42.4 → 9.6
Time: 2.7m
Precision: 64
Internal Precision: 1344
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.4326322828266477 \cdot 10^{+95}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(\frac{\frac{2 \cdot t}{x \cdot x}}{2 \cdot \sqrt{2}} - t \cdot \sqrt{2}\right) - (\left(\frac{\frac{t}{\sqrt{2}}}{x \cdot x}\right) \cdot 2 + \left(\frac{2}{\sqrt{2}} \cdot \frac{t}{x}\right))_*}\\ \mathbf{elif}\;t \le 2.6384512061600146 \cdot 10^{-276}:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt{2}} \cdot \left(t \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)\right)}{\sqrt{(2 \cdot \left(t \cdot t\right) + \left((4 \cdot \left(\frac{t}{\frac{x}{t}}\right) + \left(\frac{\ell}{\frac{x}{\ell}} \cdot 2\right))_*\right))_*}}\\ \mathbf{elif}\;t \le 6.305933245699964 \cdot 10^{-161} \lor \neg \left(t \le 5.35556584533854 \cdot 10^{+37}\right):\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{(t \cdot \left(\sqrt{2}\right) + \left(\frac{\frac{2 \cdot t}{x}}{\sqrt{2}} + \frac{2 \cdot t}{\left(x \cdot x\right) \cdot \sqrt{2}}\right))_* - \frac{\frac{2 \cdot t}{x \cdot x}}{2 \cdot \sqrt{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(t \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)\right) \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right)\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}}{\sqrt{(2 \cdot \left(t \cdot t\right) + \left((4 \cdot \left(\frac{t}{\frac{x}{t}}\right) + \left(\frac{\ell}{\frac{x}{\ell}} \cdot 2\right))_*\right))_*}}\\ \end{array}\]

Error

Bits error versus x

Bits error versus l

Bits error versus t

Derivation

  1. Split input into 4 regimes

  2. if t < -1.4326322828266477e+95

    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. Initial simplification48.8

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

    3. Taylor expanded around -inf 3.5

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

    4. Simplified3.5

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

    if -1.4326322828266477e+95 < t < 2.6384512061600146e-276

    1. Initial program 39.0

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

    2. Initial simplification39.0

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

    3. Taylor expanded around inf 17.4

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

    4. Simplified14.2

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

    6. Applied add-cube-cbrt14.2

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

    7. Applied associate-*r*14.1

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

    if 2.6384512061600146e-276 < t < 6.305933245699964e-161 or 5.35556584533854e+37 < t

    1. Initial program 47.3

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

    2. Initial simplification47.3

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

    3. Taylor expanded around inf 10.9

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

    4. Simplified10.9

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

    if 6.305933245699964e-161 < t < 5.35556584533854e+37

    1. Initial program 30.3

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

    2. Initial simplification30.3

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

    3. Taylor expanded around inf 10.6

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

    4. Simplified5.2

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

    6. Applied add-cube-cbrt5.2

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

    7. Applied associate-*r*5.1

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

    9. Applied add-cube-cbrt5.1

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

    10. Applied associate-*r*5.2

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

  4. Final simplification9.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.4326322828266477 \cdot 10^{+95}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(\frac{\frac{2 \cdot t}{x \cdot x}}{2 \cdot \sqrt{2}} - t \cdot \sqrt{2}\right) - (\left(\frac{\frac{t}{\sqrt{2}}}{x \cdot x}\right) \cdot 2 + \left(\frac{2}{\sqrt{2}} \cdot \frac{t}{x}\right))_*}\\ \mathbf{elif}\;t \le 2.6384512061600146 \cdot 10^{-276}:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt{2}} \cdot \left(t \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)\right)}{\sqrt{(2 \cdot \left(t \cdot t\right) + \left((4 \cdot \left(\frac{t}{\frac{x}{t}}\right) + \left(\frac{\ell}{\frac{x}{\ell}} \cdot 2\right))_*\right))_*}}\\ \mathbf{elif}\;t \le 6.305933245699964 \cdot 10^{-161} \lor \neg \left(t \le 5.35556584533854 \cdot 10^{+37}\right):\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{(t \cdot \left(\sqrt{2}\right) + \left(\frac{\frac{2 \cdot t}{x}}{\sqrt{2}} + \frac{2 \cdot t}{\left(x \cdot x\right) \cdot \sqrt{2}}\right))_* - \frac{\frac{2 \cdot t}{x \cdot x}}{2 \cdot \sqrt{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(t \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)\right) \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right)\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}}{\sqrt{(2 \cdot \left(t \cdot t\right) + \left((4 \cdot \left(\frac{t}{\frac{x}{t}}\right) + \left(\frac{\ell}{\frac{x}{\ell}} \cdot 2\right))_*\right))_*}}\\ \end{array}\]

Runtime

Time bar (total: 2.7m)Debug logProfile

herbie shell --seed 2018214 +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)))))