Average Error: 42.4 → 9.5
Time: 1.9m
Precision: 64
Internal Precision: 1408
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.475641717166274 \cdot 10^{+86}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(\frac{\frac{\frac{t}{x}}{x}}{\sqrt{2}} - t \cdot \sqrt{2}\right) - \left(\frac{\frac{t + t}{\sqrt{2}}}{x} + \frac{\frac{t + t}{\sqrt{2}}}{x \cdot x}\right)}\\ \mathbf{if}\;t \le 3.3413484077652453 \cdot 10^{-269}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot {t}^{2} + \left(2 \cdot \frac{\ell}{\frac{x}{\ell}} + 4 \cdot \frac{{t}^{2}}{x}\right)}}\\ \mathbf{if}\;t \le 8.234052182150638 \cdot 10^{-227}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \left(\sqrt{2} + \frac{\frac{2}{x}}{\sqrt{2}}\right) + \frac{\frac{t}{x}}{x} \cdot \left(\frac{2}{\sqrt{2}} - \frac{1}{\sqrt{2}}\right)}\\ \mathbf{if}\;t \le 1.598402083248483 \cdot 10^{+41}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot {t}^{2} + \left(2 \cdot \frac{\ell}{\frac{x}{\ell}} + 4 \cdot \frac{{t}^{2}}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \left(\sqrt{2} + \frac{\frac{2}{x}}{\sqrt{2}}\right) + \frac{\frac{t}{x}}{x} \cdot \left(\frac{2}{\sqrt{2}} - \frac{1}{\sqrt{2}}\right)}\\ \end{array}\]

Error

Bits error versus x

Bits error versus l

Bits error versus t

Derivation

  1. Split input into 3 regimes

  2. if t < -1.475641717166274e+86

    1. Initial program 48.6

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

    2. Taylor expanded around -inf 2.6

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

    3. Applied simplify2.7

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

    if -1.475641717166274e+86 < t < 3.3413484077652453e-269 or 8.234052182150638e-227 < t < 1.598402083248483e+41

    1. Initial program 38.4

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

    2. Taylor expanded around inf 17.5

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

    4. Applied unpow217.5

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

    5. Applied associate-/l*13.3

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

    if 3.3413484077652453e-269 < t < 8.234052182150638e-227 or 1.598402083248483e+41 < t

    1. Initial program 45.1

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

    2. Taylor expanded around inf 7.6

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

    3. Applied simplify7.6

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

Runtime

Time bar (total: 1.9m)Debug logProfile

herbie shell --seed '#(1070258749 1877548225 2229079127 1588002776 3179087814 1886870650)' 
(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)))))