Average Error: 42.3 → 10.2
Time: 1.7m
Precision: 64
Internal Precision: 1408
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -129247.07234099421:\\ \;\;\;\;\frac{\frac{t \cdot \sqrt{2}}{-t}}{\sqrt{2} + \frac{\frac{2}{x}}{\sqrt{2}}}\\ \mathbf{if}\;t \le -2.9216474322750386 \cdot 10^{-208}:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt{2}} \cdot \left(\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot t\right)}{\sqrt{(\left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) \cdot 2 + \left(\frac{4}{x} \cdot \left(t \cdot t\right)\right))_*}}\\ \mathbf{if}\;t \le -1.5412255347513736 \cdot 10^{-253}:\\ \;\;\;\;\frac{\frac{t \cdot \sqrt{2}}{-t}}{\sqrt{2} + \frac{\frac{2}{x}}{\sqrt{2}}}\\ \mathbf{if}\;t \le 2.002325168904788 \cdot 10^{-210}:\\ \;\;\;\;\frac{\left(t \cdot \sqrt{\sqrt{2}}\right) \cdot \sqrt{\sqrt{2}}}{\sqrt{(\left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) \cdot 2 + \left(\frac{4}{x} \cdot \left(t \cdot t\right)\right))_*}}\\ \mathbf{if}\;t \le 2.2122325362749126 \cdot 10^{-144}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{2} + \frac{\frac{2}{x}}{\sqrt{2}}}\\ \mathbf{if}\;t \le 1.4755532305137894 \cdot 10^{+143}:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt{2}} \cdot \left(\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot t\right)}{\sqrt{(\left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) \cdot 2 + \left(\frac{4}{x} \cdot \left(t \cdot t\right)\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{2} + \frac{\frac{2}{x}}{\sqrt{2}}}\\ \end{array}\]

Error

Bits error versus x

Bits error versus l

Bits error versus t

Derivation

  1. Split input into 4 regimes

  2. if t < -129247.07234099421 or -2.9216474322750386e-208 < t < -1.5412255347513736e-253

    1. Initial program 42.9

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

    2. Applied simplify42.9

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

    3. Taylor expanded around -inf 9.3

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

    4. Applied simplify9.4

      \[\leadsto \color{blue}{\frac{\frac{t \cdot \sqrt{2}}{-t}}{\frac{\frac{2}{x}}{\sqrt{2}} + \sqrt{2}}}\]

    if -129247.07234099421 < t < -2.9216474322750386e-208 or 2.2122325362749126e-144 < t < 1.4755532305137894e+143

    1. Initial program 29.2

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

    2. Applied simplify29.2

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

    3. Taylor expanded around inf 12.2

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

    4. Applied simplify7.0

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

    6. Applied add-cube-cbrt7.0

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

    7. Applied associate-*r*7.0

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

    if -1.5412255347513736e-253 < t < 2.002325168904788e-210

    1. Initial program 61.8

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

    2. Applied simplify61.8

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

    3. Taylor expanded around inf 31.5

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

    4. Applied simplify31.1

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

    6. Applied add-sqr-sqrt31.1

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

    7. Applied associate-*r*31.1

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

    if 2.002325168904788e-210 < t < 2.2122325362749126e-144 or 1.4755532305137894e+143 < t

    1. Initial program 57.9

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

    2. Applied simplify57.9

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

    3. Taylor expanded around inf 8.3

      \[\leadsto \frac{t \cdot \sqrt{2}}{\color{blue}{t \cdot \sqrt{2} + 2 \cdot \frac{t}{\sqrt{2} \cdot x}}}\]

    4. Applied simplify8.3

      \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot 1}{\frac{\frac{2}{x}}{\sqrt{2}} + \sqrt{2}}}\]
  3. Recombined 4 regimes into one program.

  4. Applied simplify10.2

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;t \le -129247.07234099421:\\ \;\;\;\;\frac{\frac{t \cdot \sqrt{2}}{-t}}{\sqrt{2} + \frac{\frac{2}{x}}{\sqrt{2}}}\\ \mathbf{if}\;t \le -2.9216474322750386 \cdot 10^{-208}:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt{2}} \cdot \left(\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot t\right)}{\sqrt{(\left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) \cdot 2 + \left(\frac{4}{x} \cdot \left(t \cdot t\right)\right))_*}}\\ \mathbf{if}\;t \le -1.5412255347513736 \cdot 10^{-253}:\\ \;\;\;\;\frac{\frac{t \cdot \sqrt{2}}{-t}}{\sqrt{2} + \frac{\frac{2}{x}}{\sqrt{2}}}\\ \mathbf{if}\;t \le 2.002325168904788 \cdot 10^{-210}:\\ \;\;\;\;\frac{\left(t \cdot \sqrt{\sqrt{2}}\right) \cdot \sqrt{\sqrt{2}}}{\sqrt{(\left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) \cdot 2 + \left(\frac{4}{x} \cdot \left(t \cdot t\right)\right))_*}}\\ \mathbf{if}\;t \le 2.2122325362749126 \cdot 10^{-144}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{2} + \frac{\frac{2}{x}}{\sqrt{2}}}\\ \mathbf{if}\;t \le 1.4755532305137894 \cdot 10^{+143}:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt{2}} \cdot \left(\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot t\right)}{\sqrt{(\left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) \cdot 2 + \left(\frac{4}{x} \cdot \left(t \cdot t\right)\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{2} + \frac{\frac{2}{x}}{\sqrt{2}}}\\ \end{array}}\]

Runtime

Time bar (total: 1.7m)Debug logProfile

herbie shell --seed '#(1070355188 2193211668 3977393919 3454156579 3755371326 1656365382)' +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)))))