Average Error: 42.2 → 9.0
Time: 2.2m
Precision: 64
Internal Precision: 1344
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -5.943549137346985 \cdot 10^{+85}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{(\left(\frac{\frac{2}{x}}{x}\right) \cdot \left(\frac{\frac{t}{2}}{\sqrt{2}} - \frac{t}{\sqrt{2}}\right) + \left((\left(\frac{2}{x}\right) \cdot \left(\frac{-t}{\sqrt{2}}\right) + \left(\sqrt{2} \cdot \left(-t\right)\right))_*\right))_*}\\ \mathbf{if}\;t \le 6.994201171459714 \cdot 10^{+55}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{(2 \cdot \left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) + \left(\left(4 \cdot t\right) \cdot \frac{t}{x}\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \left(\frac{\frac{2}{x}}{\sqrt{2}} + \sqrt{2}\right) - \frac{\frac{t}{\sqrt{2}}}{x \cdot x} \cdot \left(1 - 2\right)}\\ \end{array}\]

Error

Bits error versus x

Bits error versus l

Bits error versus t

Derivation

  1. Split input into 3 regimes

  2. if t < -5.943549137346985e+85

    1. Initial program 47.7

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

    2. Taylor expanded around -inf 3.3

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

    3. Applied simplify3.3

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

    if -5.943549137346985e+85 < t < 6.994201171459714e+55

    1. Initial program 39.1

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

    2. Taylor expanded around inf 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. Applied simplify13.6

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

    if 6.994201171459714e+55 < t

    1. Initial program 44.1

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

    2. Taylor expanded around inf 3.6

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

    3. Applied simplify3.6

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

Runtime

Time bar (total: 2.2m)Debug logProfile

herbie shell --seed 2018296 +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)))))