Average Error: 42.8 → 9.7
Time: 3.4m
Precision: 64
Internal Precision: 1408
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.0817749576247672 \cdot 10^{+126}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(\frac{\frac{\frac{t}{x}}{x}}{\sqrt{2}} - t \cdot \sqrt{2}\right) - \frac{\frac{2}{x}}{\sqrt{2}} \cdot \left(\frac{t}{x} + t\right)}\\ \mathbf{if}\;t \le -9.21963799851886 \cdot 10^{-168}:\\ \;\;\;\;\left(\sqrt[3]{\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot {t}^{2} + \left(2 \cdot \frac{\ell}{\frac{x}{\ell}} + 4 \cdot \frac{{t}^{2}}{x}\right)}}} \cdot \sqrt[3]{\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot {t}^{2} + \left(2 \cdot \frac{\ell}{\frac{x}{\ell}} + 4 \cdot \frac{{t}^{2}}{x}\right)}}}\right) \cdot \sqrt[3]{\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot {t}^{2} + \left(2 \cdot \frac{\ell}{\frac{x}{\ell}} + 4 \cdot \frac{{t}^{2}}{x}\right)}}}\\ \mathbf{if}\;t \le -2.9107726421089792 \cdot 10^{-300}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(\frac{\frac{\frac{t}{x}}{x}}{\sqrt{2}} - t \cdot \sqrt{2}\right) - \frac{\frac{2}{x}}{\sqrt{2}} \cdot \left(\frac{t}{x} + t\right)}\\ \mathbf{if}\;t \le 6.786691997666397:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{1}} \cdot \frac{t}{\sqrt{2 \cdot {t}^{2} + \left(2 \cdot \frac{\ell}{\frac{x}{\ell}} + 4 \cdot \frac{{t}^{2}}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(\frac{\frac{2}{x}}{\sqrt{2}} \cdot \left(\frac{t}{x} + t\right) + t \cdot \sqrt{2}\right) - \frac{\frac{\frac{t}{x}}{x}}{\sqrt{2}}}\\ \end{array}\]

Error

Bits error versus x

Bits error versus l

Bits error versus t

Derivation

  1. Split input into 4 regimes

  2. if t < -1.0817749576247672e+126 or -9.21963799851886e-168 < t < -2.9107726421089792e-300

    1. Initial program 57.3

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

    2. Taylor expanded around -inf 12.6

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

    3. Applied simplify12.6

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

    if -1.0817749576247672e+126 < t < -9.21963799851886e-168

    1. Initial program 25.3

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

    2. Taylor expanded around inf 10.0

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

    4. Applied unpow210.0

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

    5. Applied associate-/l*5.1

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

    7. Applied add-cube-cbrt5.0

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

    if -2.9107726421089792e-300 < t < 6.786691997666397

    1. Initial program 45.9

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

    2. Taylor expanded around inf 20.1

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

    4. Applied unpow220.1

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

    5. Applied associate-/l*15.9

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

    7. Applied *-un-lft-identity15.9

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

    8. Applied sqrt-prod15.9

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

    9. Applied times-frac15.9

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

    if 6.786691997666397 < t

    1. Initial program 41.2

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

    2. Taylor expanded around inf 5.7

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

    3. Applied simplify5.7

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

Runtime

Time bar (total: 3.4m)Debug logProfile

herbie shell --seed '#(1063027428 1192549564 1443466578 604016274 3637110559 1698629644)' 
(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)))))