Average Error: 42.4 → 8.8
Time: 1.7m
Precision: 64
Internal Precision: 1408
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -4.5299787915372384 \cdot 10^{+122}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(\frac{\frac{\frac{t}{x}}{x}}{\sqrt{2}} - t \cdot \sqrt{2}\right) - \left(\frac{\frac{t + t}{\sqrt{2}}}{x} + \frac{\frac{t + t}{\sqrt{2}}}{x \cdot x}\right)}\\ \mathbf{if}\;t \le -4.4287023113340517 \cdot 10^{-159}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot t\right)}{\sqrt{2 \cdot {t}^{2} + \left(2 \cdot \frac{\ell}{\frac{x}{\ell}} + 4 \cdot \frac{{t}^{2}}{x}\right)}}\\ \mathbf{if}\;t \le -1.4188303149527196 \cdot 10^{-260}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(\frac{\frac{\frac{t}{x}}{x}}{\sqrt{2}} - t \cdot \sqrt{2}\right) - \left(\frac{\frac{t + t}{\sqrt{2}}}{x} + \frac{\frac{t + t}{\sqrt{2}}}{x \cdot x}\right)}\\ \mathbf{if}\;t \le 7.257921404658838 \cdot 10^{+111}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot t\right)}{\sqrt{2 \cdot {t}^{2} + \left(2 \cdot \frac{\ell}{\frac{x}{\ell}} + 4 \cdot \frac{{t}^{2}}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \left(\sqrt{2} + \frac{\frac{2}{x}}{\sqrt{2}}\right) + \frac{\frac{t}{x}}{x} \cdot \left(\frac{2}{\sqrt{2}} - \frac{1}{\sqrt{2}}\right)}\\ \end{array}\]

Error

Bits error versus x

Bits error versus l

Bits error versus t

Derivation

  1. Split input into 3 regimes

  2. if t < -4.5299787915372384e+122 or -4.4287023113340517e-159 < t < -1.4188303149527196e-260

    1. Initial program 56.5

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

    2. Taylor expanded around -inf 11.0

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

    3. Applied simplify11.0

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

    if -4.5299787915372384e+122 < t < -4.4287023113340517e-159 or -1.4188303149527196e-260 < t < 7.257921404658838e+111

    1. Initial program 33.0

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

    2. Taylor expanded around inf 14.7

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

    4. Applied unpow214.7

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

    5. Applied associate-/l*10.0

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

    7. Applied add-cube-cbrt10.0

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

    8. Applied associate-*l*10.0

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

    if 7.257921404658838e+111 < 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 2.6

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

    3. Applied simplify2.6

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

Runtime

Time bar (total: 1.7m)Debug logProfile

herbie shell --seed '#(1064173506 2580572819 2847706409 4129882574 1125180799 1845288547)' 
(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)))))