Average Error: 42.2 → 9.3
Time: 1.9m
Precision: 64
Internal Precision: 1344
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.2374660040310888 \cdot 10^{+101}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{(\left(\frac{\frac{t}{x}}{x}\right) \cdot \left(\frac{1}{\sqrt{2}}\right) + \left(\left(-t\right) \cdot \sqrt{2}\right))_* - \left(t + \frac{t}{x}\right) \cdot \frac{\frac{2}{x}}{\sqrt{2}}}\\ \mathbf{elif}\;t \le -1.5374680613088642 \cdot 10^{-143}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(t \cdot \sqrt[3]{\sqrt{2}}\right)}{\sqrt{(2 \cdot \left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) + \left(\left(t \cdot 4\right) \cdot \frac{t}{x}\right))_*}}\\ \mathbf{elif}\;t \le -5.5047625092146135 \cdot 10^{-182}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{(\left(\frac{\frac{t}{x}}{x}\right) \cdot \left(\frac{1}{\sqrt{2}}\right) + \left(\left(-t\right) \cdot \sqrt{2}\right))_* - \left(t + \frac{t}{x}\right) \cdot \frac{\frac{2}{x}}{\sqrt{2}}}\\ \mathbf{elif}\;t \le 2.2186899003450603 \cdot 10^{+102}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(t \cdot \sqrt[3]{\sqrt{2}}\right)}{\sqrt{(2 \cdot \left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) + \left(\left(t \cdot 4\right) \cdot \frac{t}{x}\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{(\left(\frac{t}{\sqrt{2}}\right) \cdot \left(\frac{\frac{2}{x}}{x} + \frac{2}{x}\right) + \left((\left(\frac{1}{\sqrt{2}}\right) \cdot \left(\frac{-t}{x \cdot x}\right) + \left(t \cdot \sqrt{2}\right))_*\right))_*}\\ \end{array}\]

Error

Bits error versus x

Bits error versus l

Bits error versus t

Derivation

  1. Split input into 3 regimes

  2. if t < -1.2374660040310888e+101 or -1.5374680613088642e-143 < t < -5.5047625092146135e-182

    1. Initial program 50.4

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

    2. Taylor expanded around -inf 5.6

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

    3. Simplified5.6

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

    if -1.2374660040310888e+101 < t < -1.5374680613088642e-143 or -5.5047625092146135e-182 < t < 2.2186899003450603e+102

    1. Initial program 36.5

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

    2. Taylor expanded around inf 17.2

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

    3. Simplified12.9

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

    5. Applied add-cube-cbrt12.9

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

    6. Applied associate-*l*12.9

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

    if 2.2186899003450603e+102 < t

    1. Initial program 49.6

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

    2. Taylor expanded around inf 2.9

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

    3. Simplified2.9

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

  4. Final simplification9.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.2374660040310888 \cdot 10^{+101}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{(\left(\frac{\frac{t}{x}}{x}\right) \cdot \left(\frac{1}{\sqrt{2}}\right) + \left(\left(-t\right) \cdot \sqrt{2}\right))_* - \left(t + \frac{t}{x}\right) \cdot \frac{\frac{2}{x}}{\sqrt{2}}}\\ \mathbf{elif}\;t \le -1.5374680613088642 \cdot 10^{-143}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(t \cdot \sqrt[3]{\sqrt{2}}\right)}{\sqrt{(2 \cdot \left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) + \left(\left(t \cdot 4\right) \cdot \frac{t}{x}\right))_*}}\\ \mathbf{elif}\;t \le -5.5047625092146135 \cdot 10^{-182}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{(\left(\frac{\frac{t}{x}}{x}\right) \cdot \left(\frac{1}{\sqrt{2}}\right) + \left(\left(-t\right) \cdot \sqrt{2}\right))_* - \left(t + \frac{t}{x}\right) \cdot \frac{\frac{2}{x}}{\sqrt{2}}}\\ \mathbf{elif}\;t \le 2.2186899003450603 \cdot 10^{+102}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(t \cdot \sqrt[3]{\sqrt{2}}\right)}{\sqrt{(2 \cdot \left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) + \left(\left(t \cdot 4\right) \cdot \frac{t}{x}\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{(\left(\frac{t}{\sqrt{2}}\right) \cdot \left(\frac{\frac{2}{x}}{x} + \frac{2}{x}\right) + \left((\left(\frac{1}{\sqrt{2}}\right) \cdot \left(\frac{-t}{x \cdot x}\right) + \left(t \cdot \sqrt{2}\right))_*\right))_*}\\ \end{array}\]

Runtime

Time bar (total: 1.9m)Debug logProfile

herbie shell --seed 2018215 +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)))))