Average Error: 42.4 → 8.9
Time: 3.6m
Precision: 64
Internal Precision: 1344
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -4.904779183137194 \cdot 10^{+127}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(1 - 2\right) \cdot \frac{\frac{t}{\sqrt{2}}}{x \cdot x} - \left(\frac{\frac{2}{x}}{\sqrt{2}} + \sqrt{2}\right) \cdot t}\\ \mathbf{if}\;t \le 3.007086639531952 \cdot 10^{-199} \lor \neg \left(t \le 1.0823464268273412 \cdot 10^{-162} \lor \neg \left(t \le 4.3018340931781265 \cdot 10^{+69}\right)\right):\\ \;\;\;\;\left(\sqrt[3]{\frac{t \cdot \sqrt{2}}{\sqrt{\frac{\ell \cdot 2}{\frac{x}{\ell}} + \left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)}}} \cdot \sqrt[3]{\frac{t \cdot \sqrt{2}}{\sqrt{\frac{\ell \cdot 2}{\frac{x}{\ell}} + \left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)}}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{\frac{t \cdot \sqrt{2}}{\sqrt{\frac{\ell \cdot 2}{\frac{x}{\ell}} + \left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)}}}} \cdot \sqrt[3]{\sqrt[3]{\frac{t \cdot \sqrt{2}}{\sqrt{\frac{\ell \cdot 2}{\frac{x}{\ell}} + \left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)}}} \cdot \sqrt[3]{\frac{t \cdot \sqrt{2}}{\sqrt{\frac{\ell \cdot 2}{\frac{x}{\ell}} + \left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\frac{t}{\left(x \cdot x\right) \cdot \sqrt{2}} \cdot \left(2 - 1\right) + \left(\frac{\frac{2}{x}}{\sqrt{2}} + \sqrt{2}\right) \cdot t}\\ \end{array}\]

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 3 regimes

  2. if t < -4.904779183137194e+127

    1. Initial program 55.8

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

    2. Taylor expanded around -inf 2.1

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

    3. Applied simplify2.1

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

    if -4.904779183137194e+127 < t < 3.007086639531952e-199 or 1.0823464268273412e-162 < t < 4.3018340931781265e+69

    1. Initial program 36.2

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

    2. Taylor expanded around inf 16.3

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

    3. Applied simplify12.2

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

    5. Applied add-cube-cbrt12.2

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

    7. Applied add-cube-cbrt12.2

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

    8. Applied cbrt-prod12.3

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

    if 3.007086639531952e-199 < t < 1.0823464268273412e-162 or 4.3018340931781265e+69 < t

    1. Initial program 46.7

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

    2. Taylor expanded around inf 6.3

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

    3. Applied simplify6.3

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

  4. Applied simplify8.9

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;t \le -4.904779183137194 \cdot 10^{+127}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(1 - 2\right) \cdot \frac{\frac{t}{\sqrt{2}}}{x \cdot x} - \left(\frac{\frac{2}{x}}{\sqrt{2}} + \sqrt{2}\right) \cdot t}\\ \mathbf{if}\;t \le 3.007086639531952 \cdot 10^{-199} \lor \neg \left(t \le 1.0823464268273412 \cdot 10^{-162} \lor \neg \left(t \le 4.3018340931781265 \cdot 10^{+69}\right)\right):\\ \;\;\;\;\left(\sqrt[3]{\frac{t \cdot \sqrt{2}}{\sqrt{\frac{\ell \cdot 2}{\frac{x}{\ell}} + \left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)}}} \cdot \sqrt[3]{\frac{t \cdot \sqrt{2}}{\sqrt{\frac{\ell \cdot 2}{\frac{x}{\ell}} + \left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)}}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{\frac{t \cdot \sqrt{2}}{\sqrt{\frac{\ell \cdot 2}{\frac{x}{\ell}} + \left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)}}}} \cdot \sqrt[3]{\sqrt[3]{\frac{t \cdot \sqrt{2}}{\sqrt{\frac{\ell \cdot 2}{\frac{x}{\ell}} + \left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)}}} \cdot \sqrt[3]{\frac{t \cdot \sqrt{2}}{\sqrt{\frac{\ell \cdot 2}{\frac{x}{\ell}} + \left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\frac{t}{\left(x \cdot x\right) \cdot \sqrt{2}} \cdot \left(2 - 1\right) + \left(\frac{\frac{2}{x}}{\sqrt{2}} + \sqrt{2}\right) \cdot t}\\ \end{array}}\]

Runtime

Time bar (total: 3.6m)Debug logProfile

herbie shell --seed 2018178 
(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)))))