Average Error: 52.9 → 17.2
Time: 2.3m
Precision: 64
Ground Truth: 128
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left({\ell}^2 + 2 \cdot {t}^2\right) - {\ell}^2}}\]
⬇
\[\begin{array}{l} \mathbf{if}\;t \le -4.668953112360352 \cdot 10^{+47}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(\frac{\frac{\frac{t}{x}}{\sqrt{2}}}{x} - \frac{2 + \frac{2}{x}}{\frac{\sqrt{2}}{\frac{t}{x}}}\right) - \sqrt{2} \cdot t}\\ \mathbf{if}\;t \le 4.630417065006707 \cdot 10^{-299}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{{\left(\sqrt{\sqrt{2 \cdot {t}^2 + \left(2 \cdot \frac{\ell}{\frac{x}{\ell}} + 4 \cdot \frac{{t}^2}{x}\right)}}\right)}^2}\\ \mathbf{if}\;t \le 2.584703619196454 \cdot 10^{-179}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(\frac{\frac{2}{x}}{\sqrt{2}} \cdot \left(\frac{t}{x} + t\right) - \frac{t}{{x}^2 \cdot \sqrt{2}}\right) + \sqrt{2} \cdot t}\\ \mathbf{if}\;t \le 2.9674506545823876 \cdot 10^{-06}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot {t}^2 + \left(2 \cdot \frac{\ell}{\frac{x}{\ell}} + 4 \cdot \frac{{t}^2}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(\frac{\frac{2}{x}}{\sqrt{2}} \cdot \left(\frac{t}{x} + t\right) - \frac{t}{{x}^2 \cdot \sqrt{2}}\right) + \sqrt{2} \cdot t}\\ \end{array}\]

Error

Bits error versus x

Bits error versus l

Bits error versus t

Derivation

  1. Split input into 4 regimes.

  2. if t < -4.668953112360352e+47

    1. Initial program 54.3

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

    2. Applied taylor 24.2

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

    3. Taylor expanded around -inf 24.2

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

    4. Applied simplify 24.3

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

    5. Applied simplify 24.3

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

    if -4.668953112360352e+47 < t < 4.630417065006707e-299

    1. Initial program 50.1

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

    2. Applied taylor 14.6

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

    3. Taylor expanded around inf 14.6

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

    5. Applied square-mult 14.6

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

    6. Applied associate-/l* 12.1

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

    8. Applied add-sqr-sqrt 12.3

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

    if 4.630417065006707e-299 < t < 2.584703619196454e-179 or 2.9674506545823876e-06 < t

    1. Initial program 55.0

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

    2. Applied taylor 21.2

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

    3. Taylor expanded around inf 21.2

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

    4. Applied simplify 17.8

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

    5. Applied simplify 17.8

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

    if 2.584703619196454e-179 < t < 2.9674506545823876e-06

    1. Initial program 46.6

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

    2. Applied taylor 11.3

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

    3. Taylor expanded around inf 11.3

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

    5. Applied square-mult 11.3

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

    6. Applied associate-/l* 8.3

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

Runtime

Total time: 2.3m Debug log

Please include this information when filing a bug report:

herbie --seed '#(2798093225 3646344418 2008128416 4149615316 3565459355 3721252699)'
(FPCore (x l t)
  :name "Toniolo and Linder, Equation (7)"
  (/ (* (sqrt 2) t) (sqrt (- (* (/ (+ x 1) (- x 1)) (+ (sqr l) (* 2 (sqr t)))) (sqr l)))))