Average Error: 42.4 → 10.0
Time: 1.5m
Precision: 64
Internal Precision: 1408
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -5.208116910601954 \cdot 10^{+139}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\frac{\frac{t}{x}}{x} \cdot \frac{1}{\sqrt{2}} - (\left(\frac{2}{\sqrt{2}}\right) \cdot \left(\frac{\frac{t}{x}}{x}\right) + \left((\left(\frac{2}{\sqrt{2}}\right) \cdot \left(\frac{t}{x}\right) + \left(t \cdot \sqrt{2}\right))_*\right))_*}\\ \mathbf{if}\;t \le -9.84381717448642 \cdot 10^{-131}:\\ \;\;\;\;\frac{\left(\left(t \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)\right) \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right)\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}}{\sqrt{(2 \cdot \left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) + \left(\frac{4}{x} \cdot \left(t \cdot t\right)\right))_*}}\\ \mathbf{if}\;t \le -6.432140854570408 \cdot 10^{-257}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\frac{\frac{t}{x}}{x} \cdot \frac{1}{\sqrt{2}} - (\left(\frac{2}{\sqrt{2}}\right) \cdot \left(\frac{\frac{t}{x}}{x}\right) + \left((\left(\frac{2}{\sqrt{2}}\right) \cdot \left(\frac{t}{x}\right) + \left(t \cdot \sqrt{2}\right))_*\right))_*}\\ \mathbf{if}\;t \le 9.803892206344536 \cdot 10^{+115}:\\ \;\;\;\;\frac{\left(\left(t \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)\right) \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right)\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}}{\sqrt{(2 \cdot \left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) + \left(\frac{4}{x} \cdot \left(t \cdot t\right)\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\frac{\frac{t}{x}}{x} \cdot \left(\frac{2}{\sqrt{2}} - \frac{1}{\sqrt{2}}\right) + (\left(\frac{t}{\sqrt{2}}\right) \cdot \left(\frac{2}{x}\right) + \left(t \cdot \sqrt{2}\right))_*}\\ \end{array}\]

Error

Bits error versus x

Bits error versus l

Bits error versus t

Derivation

  1. Split input into 3 regimes

  2. if t < -5.208116910601954e+139 or -9.84381717448642e-131 < t < -6.432140854570408e-257

    1. Initial program 57.7

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

    2. Taylor expanded around -inf 12.5

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

    3. Applied simplify12.5

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

    if -5.208116910601954e+139 < t < -9.84381717448642e-131 or -6.432140854570408e-257 < t < 9.803892206344536e+115

    1. Initial program 32.5

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

    2. Taylor expanded around inf 15.8

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

    3. Applied simplify11.3

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

    5. Applied add-cube-cbrt11.3

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

    6. Applied associate-*r*11.3

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

    8. Applied add-cube-cbrt11.3

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

    9. Applied associate-*r*11.3

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

    if 9.803892206344536e+115 < t

    1. Initial program 52.5

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

    2. Taylor expanded around inf 2.8

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

    3. Applied simplify2.8

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

Runtime

Time bar (total: 1.5m)Debug logProfile

herbie shell --seed '#(1064269945 2896236262 301053905 1701069080 1701464310 1614783279)' +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)))))