Average Error: 42.8 → 27.2
Time: 1.0m
Precision: 64
Internal Precision: 1344
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.8500924360896874 \cdot 10^{-215} \lor \neg \left(t \le 9.869471449699305 \cdot 10^{-188}\right):\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\left(t \cdot t\right) \cdot \left(2 + \frac{4}{x}\right) + \left(2 \cdot \ell\right) \cdot \frac{\ell}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\frac{\sqrt{\left(\left(\left(t \cdot t\right) \cdot x\right) \cdot \left(2 + \frac{4}{x}\right) + \left(2 \cdot \ell\right) \cdot \ell\right) \cdot \left(2 - \frac{4}{x}\right)}}{\sqrt{\left(2 - \frac{4}{x}\right) \cdot x}}}\\ \end{array}\]

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 2 regimes

  2. if t < -1.8500924360896874e-215 or 9.869471449699305e-188 < t

    1. Initial program 39.9

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

    2. Initial simplification39.9

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

    3. Taylor expanded around -inf 31.1

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

    4. Simplified27.4

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

    if -1.8500924360896874e-215 < t < 9.869471449699305e-188

    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. Initial simplification61.8

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

    3. Taylor expanded around -inf 30.9

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

    4. Simplified29.8

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

    6. Applied flip-+29.8

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

    7. Applied associate-*l/29.8

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

    8. Applied associate-*l/30.9

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

    9. Applied frac-add31.2

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

    10. Applied sqrt-div26.2

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

    11. Simplified26.2

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

  4. Final simplification27.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.8500924360896874 \cdot 10^{-215} \lor \neg \left(t \le 9.869471449699305 \cdot 10^{-188}\right):\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\left(t \cdot t\right) \cdot \left(2 + \frac{4}{x}\right) + \left(2 \cdot \ell\right) \cdot \frac{\ell}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\frac{\sqrt{\left(\left(\left(t \cdot t\right) \cdot x\right) \cdot \left(2 + \frac{4}{x}\right) + \left(2 \cdot \ell\right) \cdot \ell\right) \cdot \left(2 - \frac{4}{x}\right)}}{\sqrt{\left(2 - \frac{4}{x}\right) \cdot x}}}\\ \end{array}\]

Runtime

Time bar (total: 1.0m)Debug logProfile

herbie shell --seed 2018248 
(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)))))