Average Error: 42.3 → 9.2
Time: 2.3m
Precision: 64
Internal Precision: 1344
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -4.1325673111713254 \cdot 10^{+89}:\\ \;\;\;\;\frac{t}{\left(-t\right) - \frac{t}{2} \cdot \frac{2}{x}}\\ \mathbf{if}\;t \le 1.9105398178236134 \cdot 10^{+115}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{(\left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) \cdot 2 + \left(\left(t \cdot t\right) \cdot \frac{4}{x}\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\frac{2}{x}}{\sqrt{2}} + \sqrt{2}}\\ \end{array}\]

Error

Bits error versus x

Bits error versus l

Bits error versus t

Derivation

  1. Split input into 3 regimes

  2. if t < -4.1325673111713254e+89

    1. Initial program 48.3

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

    2. Applied simplify48.3

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

    3. Taylor expanded around -inf 3.5

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

    4. Applied simplify3.5

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

    if -4.1325673111713254e+89 < t < 1.9105398178236134e+115

    1. Initial program 37.0

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

    2. Applied simplify37.0

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

    3. Taylor expanded around inf 17.7

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

    4. Applied simplify13.2

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

    if 1.9105398178236134e+115 < t

    1. Initial program 53.0

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

    2. Applied simplify53.0

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

    3. Taylor expanded around inf 2.6

      \[\leadsto \frac{t \cdot \sqrt{2}}{\color{blue}{t \cdot \sqrt{2} + 2 \cdot \frac{t}{\sqrt{2} \cdot x}}}\]

    4. Applied simplify2.5

      \[\leadsto \color{blue}{\frac{1 \cdot \sqrt{2}}{\frac{\frac{2}{x}}{\sqrt{2}} + \sqrt{2}}}\]
  3. Recombined 3 regimes into one program.

  4. Applied simplify9.2

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;t \le -4.1325673111713254 \cdot 10^{+89}:\\ \;\;\;\;\frac{t}{\left(-t\right) - \frac{t}{2} \cdot \frac{2}{x}}\\ \mathbf{if}\;t \le 1.9105398178236134 \cdot 10^{+115}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{(\left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) \cdot 2 + \left(\left(t \cdot t\right) \cdot \frac{4}{x}\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\frac{2}{x}}{\sqrt{2}} + \sqrt{2}}\\ \end{array}}\]

Runtime

Time bar (total: 2.3m)Debug logProfile

herbie shell --seed 2018207 +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)))))