Average Error: 42.2 → 9.1
Time: 34.7s
Precision: 64
Internal Precision: 128
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -7.04804877865123 \cdot 10^{+57}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left|(\left(\frac{-2}{x}\right) \cdot \left(\frac{t}{\sqrt{2}}\right) + \left(\frac{\frac{\frac{t}{\sqrt{2}}}{x}}{x} - \sqrt{2} \cdot t\right))_*\right|}\\ \mathbf{elif}\;t \le 1.1295388211435537 \cdot 10^{+54}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left|\sqrt{(\left(t \cdot t\right) \cdot \left(\frac{4}{x}\right) + \left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_* \cdot 2\right))_*}\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\frac{\frac{2}{x}}{x} \cdot \left(\frac{t}{\sqrt{2}} - \frac{\frac{t}{\sqrt{2}}}{2}\right) + (\left(\frac{t}{x \cdot \sqrt{2}}\right) \cdot 2 + \left(\sqrt{2} \cdot t\right))_*}\\ \end{array}\]

Error

Bits error versus x

Bits error versus l

Bits error versus t

Derivation

  1. Split input into 3 regimes

  2. if t < -7.04804877865123e+57

    1. Initial program 45.7

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

    2. Taylor expanded around -inf 44.6

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

    3. Simplified44.2

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

    5. Applied add-sqr-sqrt44.2

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

    6. Applied rem-sqrt-square44.2

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

    7. Simplified42.0

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

    8. Taylor expanded around -inf 4.9

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

    9. Simplified4.9

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

    if -7.04804877865123e+57 < t < 1.1295388211435537e+54

    1. Initial program 39.7

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

    2. Taylor expanded around -inf 17.7

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

    3. Simplified17.7

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

    5. Applied add-sqr-sqrt17.7

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

    6. Applied rem-sqrt-square17.7

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

    7. Simplified13.7

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

    8. Taylor expanded around -inf 13.7

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

    if 1.1295388211435537e+54 < t

    1. Initial program 43.9

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

    2. Taylor expanded around inf 3.9

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

    3. Simplified3.9

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

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -7.04804877865123 \cdot 10^{+57}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left|(\left(\frac{-2}{x}\right) \cdot \left(\frac{t}{\sqrt{2}}\right) + \left(\frac{\frac{\frac{t}{\sqrt{2}}}{x}}{x} - \sqrt{2} \cdot t\right))_*\right|}\\ \mathbf{elif}\;t \le 1.1295388211435537 \cdot 10^{+54}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left|\sqrt{(\left(t \cdot t\right) \cdot \left(\frac{4}{x}\right) + \left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_* \cdot 2\right))_*}\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\frac{\frac{2}{x}}{x} \cdot \left(\frac{t}{\sqrt{2}} - \frac{\frac{t}{\sqrt{2}}}{2}\right) + (\left(\frac{t}{x \cdot \sqrt{2}}\right) \cdot 2 + \left(\sqrt{2} \cdot t\right))_*}\\ \end{array}\]

Reproduce

herbie shell --seed 2019089 +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)))))