Average Error: 42.7 → 9.1
Time: 1.0m
Precision: 64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -4.57263411279645 \cdot 10^{+130}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\frac{t}{x \cdot x} \cdot \frac{1}{\sqrt{2}} - \left(\frac{2}{\sqrt{2}} \cdot \left(\frac{t}{x} + \frac{t}{x \cdot x}\right) + \sqrt{2} \cdot t\right)}\\ \mathbf{elif}\;t \le 1.0219528248363 \cdot 10^{-246}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot t\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)}{\sqrt{\left(\ell \cdot 2\right) \cdot \frac{\ell}{x} + \left(\frac{4}{x} + 2\right) \cdot \left(t \cdot t\right)}}\\ \mathbf{elif}\;t \le 1.2481357721194104 \cdot 10^{-157}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\sqrt{2} \cdot t - \frac{t}{x \cdot x} \cdot \frac{1}{\sqrt{2}}\right) + \left(\frac{\frac{2}{x}}{x} + \frac{2}{x}\right) \cdot \frac{t}{\sqrt{2}}}\\ \mathbf{elif}\;t \le 3.841014328450401 \cdot 10^{+94}:\\ \;\;\;\;\frac{\left(t \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)\right)}{\sqrt{\left(\ell \cdot 2\right) \cdot \frac{\ell}{x} + \left(\frac{4}{x} + 2\right) \cdot \left(t \cdot t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\sqrt{2} \cdot t - \frac{t}{x \cdot x} \cdot \frac{1}{\sqrt{2}}\right) + \left(\frac{\frac{2}{x}}{x} + \frac{2}{x}\right) \cdot \frac{t}{\sqrt{2}}}\\ \end{array}\]

Error

Bits error versus x

Bits error versus l

Bits error versus t

Derivation

  1. Split input into 4 regimes

  2. if t < -4.57263411279645e+130

    1. Initial program 56.4

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

    2. Taylor expanded around -inf 2.0

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

    3. Simplified2.0

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

    if -4.57263411279645e+130 < t < 1.0219528248363e-246

    1. Initial program 37.5

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

    2. Taylor expanded around inf 17.8

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

    3. Simplified14.1

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

    5. Applied add-cube-cbrt14.1

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

    6. Applied associate-*l*14.1

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

    if 1.0219528248363e-246 < t < 1.2481357721194104e-157 or 3.841014328450401e+94 < t

    1. Initial program 52.2

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

    2. Taylor expanded around inf 10.2

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

    3. Simplified10.2

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

    if 1.2481357721194104e-157 < t < 3.841014328450401e+94

    1. Initial program 27.3

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

    2. Taylor expanded around inf 10.2

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

    3. Simplified4.9

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

    5. Applied add-cube-cbrt4.9

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

    6. Applied associate-*l*4.8

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

    8. Applied add-cube-cbrt4.8

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

    9. Applied associate-*l*4.9

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

    11. Applied associate-*r*4.9

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

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -4.57263411279645 \cdot 10^{+130}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\frac{t}{x \cdot x} \cdot \frac{1}{\sqrt{2}} - \left(\frac{2}{\sqrt{2}} \cdot \left(\frac{t}{x} + \frac{t}{x \cdot x}\right) + \sqrt{2} \cdot t\right)}\\ \mathbf{elif}\;t \le 1.0219528248363 \cdot 10^{-246}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot t\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)}{\sqrt{\left(\ell \cdot 2\right) \cdot \frac{\ell}{x} + \left(\frac{4}{x} + 2\right) \cdot \left(t \cdot t\right)}}\\ \mathbf{elif}\;t \le 1.2481357721194104 \cdot 10^{-157}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\sqrt{2} \cdot t - \frac{t}{x \cdot x} \cdot \frac{1}{\sqrt{2}}\right) + \left(\frac{\frac{2}{x}}{x} + \frac{2}{x}\right) \cdot \frac{t}{\sqrt{2}}}\\ \mathbf{elif}\;t \le 3.841014328450401 \cdot 10^{+94}:\\ \;\;\;\;\frac{\left(t \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)\right)}{\sqrt{\left(\ell \cdot 2\right) \cdot \frac{\ell}{x} + \left(\frac{4}{x} + 2\right) \cdot \left(t \cdot t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\sqrt{2} \cdot t - \frac{t}{x \cdot x} \cdot \frac{1}{\sqrt{2}}\right) + \left(\frac{\frac{2}{x}}{x} + \frac{2}{x}\right) \cdot \frac{t}{\sqrt{2}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019100 
(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)))))