Average Error: 42.5 → 8.8
Time: 2.6m
Precision: 64
Internal Precision: 1344
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -6.770263660128096 \cdot 10^{+86}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(1 - 2\right) \cdot \frac{\frac{t}{\sqrt{2}}}{x \cdot x} - \left(\frac{\frac{2}{x}}{\sqrt{2}} + \sqrt{2}\right) \cdot t}\\ \mathbf{if}\;t \le -1.5690091250953074 \cdot 10^{-161}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot 2}{\frac{x}{\ell}} + \left(t \cdot t\right) \cdot \left(2 + \frac{4}{x}\right)}} \cdot \frac{t}{\sqrt{1}}\\ \mathbf{if}\;t \le -2.6516558429843984 \cdot 10^{-252}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(1 - 2\right) \cdot \frac{\frac{t}{\sqrt{2}}}{x \cdot x} - \left(\frac{\frac{2}{x}}{\sqrt{2}} + \sqrt{2}\right) \cdot t}\\ \mathbf{if}\;t \le 5.6987767492158925 \cdot 10^{-233}:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt{2}} \cdot \left(\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot t\right)}{\sqrt{\frac{\ell \cdot 2}{\frac{x}{\ell}} + \left(t \cdot t\right) \cdot \left(2 + \frac{4}{x}\right)}}\\ \mathbf{if}\;t \le 3.525145520712021 \cdot 10^{-191} \lor \neg \left(t \le 9.81576299589015 \cdot 10^{+66}\right):\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(\frac{\frac{2}{x}}{\sqrt{2}} + \sqrt{2}\right) \cdot t + \frac{t}{\left(x \cdot x\right) \cdot \sqrt{2}} \cdot \left(2 - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot 2}{\frac{x}{\ell}} + \left(t \cdot t\right) \cdot \left(2 + \frac{4}{x}\right)}} \cdot \frac{t}{\sqrt{1}}\\ \end{array}\]

Error

Bits error versus x

Bits error versus l

Bits error versus t

Derivation

  1. Split input into 4 regimes

  2. if t < -6.770263660128096e+86 or -1.5690091250953074e-161 < t < -2.6516558429843984e-252

    1. Initial program 50.9

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

    2. Taylor expanded around -inf 9.0

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

    3. Applied simplify9.0

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

    if -6.770263660128096e+86 < t < -1.5690091250953074e-161 or 3.525145520712021e-191 < t < 9.81576299589015e+66

    1. Initial program 29.5

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

    2. Taylor expanded around inf 11.1

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

    3. Applied simplify6.3

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

    5. Applied *-un-lft-identity6.3

      \[\leadsto \frac{t \cdot \sqrt{2}}{\sqrt{\color{blue}{1 \cdot \left(\left(\frac{4}{x} + 2\right) \cdot \left(t \cdot t\right) + \frac{2 \cdot \ell}{\frac{x}{\ell}}\right)}}}\]

    6. Applied sqrt-prod6.3

      \[\leadsto \frac{t \cdot \sqrt{2}}{\color{blue}{\sqrt{1} \cdot \sqrt{\left(\frac{4}{x} + 2\right) \cdot \left(t \cdot t\right) + \frac{2 \cdot \ell}{\frac{x}{\ell}}}}}\]

    7. Applied times-frac6.2

      \[\leadsto \color{blue}{\frac{t}{\sqrt{1}} \cdot \frac{\sqrt{2}}{\sqrt{\left(\frac{4}{x} + 2\right) \cdot \left(t \cdot t\right) + \frac{2 \cdot \ell}{\frac{x}{\ell}}}}}\]

    if -2.6516558429843984e-252 < t < 5.6987767492158925e-233

    1. Initial program 61.6

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

    2. Taylor expanded around inf 28.8

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

    3. Applied simplify28.2

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

    5. Applied add-cube-cbrt28.2

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

    6. Applied associate-*r*28.2

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

    if 5.6987767492158925e-233 < t < 3.525145520712021e-191 or 9.81576299589015e+66 < t

    1. Initial program 47.2

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

    2. Taylor expanded around inf 6.5

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

    3. Applied simplify6.5

      \[\leadsto \color{blue}{\frac{t \cdot \sqrt{2}}{t \cdot \left(\sqrt{2} + \frac{\frac{2}{x}}{\sqrt{2}}\right) + \frac{t}{\left(x \cdot x\right) \cdot \sqrt{2}} \cdot \left(2 - 1\right)}}\]
  3. Recombined 4 regimes into one program.

  4. Applied simplify8.8

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;t \le -6.770263660128096 \cdot 10^{+86}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(1 - 2\right) \cdot \frac{\frac{t}{\sqrt{2}}}{x \cdot x} - \left(\frac{\frac{2}{x}}{\sqrt{2}} + \sqrt{2}\right) \cdot t}\\ \mathbf{if}\;t \le -1.5690091250953074 \cdot 10^{-161}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot 2}{\frac{x}{\ell}} + \left(t \cdot t\right) \cdot \left(2 + \frac{4}{x}\right)}} \cdot \frac{t}{\sqrt{1}}\\ \mathbf{if}\;t \le -2.6516558429843984 \cdot 10^{-252}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(1 - 2\right) \cdot \frac{\frac{t}{\sqrt{2}}}{x \cdot x} - \left(\frac{\frac{2}{x}}{\sqrt{2}} + \sqrt{2}\right) \cdot t}\\ \mathbf{if}\;t \le 5.6987767492158925 \cdot 10^{-233}:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt{2}} \cdot \left(\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot t\right)}{\sqrt{\frac{\ell \cdot 2}{\frac{x}{\ell}} + \left(t \cdot t\right) \cdot \left(2 + \frac{4}{x}\right)}}\\ \mathbf{if}\;t \le 3.525145520712021 \cdot 10^{-191} \lor \neg \left(t \le 9.81576299589015 \cdot 10^{+66}\right):\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(\frac{\frac{2}{x}}{\sqrt{2}} + \sqrt{2}\right) \cdot t + \frac{t}{\left(x \cdot x\right) \cdot \sqrt{2}} \cdot \left(2 - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot 2}{\frac{x}{\ell}} + \left(t \cdot t\right) \cdot \left(2 + \frac{4}{x}\right)}} \cdot \frac{t}{\sqrt{1}}\\ \end{array}}\]

Runtime

Time bar (total: 2.6m)Debug logProfile

herbie shell --seed '#(1071501266 3581234924 1086666455 2685055582 1243441566 1802958749)' 
(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)))))