Average Error: 42.5 → 9.0
Time: 2.0m
Precision: 64
Internal Precision: 1344
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -6.601388653510457 \cdot 10^{+138}:\\ \;\;\;\;\frac{t}{\left(-t\right) - \frac{t}{2} \cdot \frac{2}{x}}\\ \mathbf{if}\;t \le 5.477946837407569 \cdot 10^{+42}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{(\left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) \cdot 2 + \left(\left(t \cdot t\right) \cdot \frac{4}{x}\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\frac{2}{x}}{\sqrt{2}} + \sqrt{2}}\\ \end{array}\]

Error

Bits error versus x

Bits error versus l

Bits error versus t

Derivation

  1. Split input into 3 regimes

  2. if t < -6.601388653510457e+138

    1. Initial program 58.1

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

    2. Applied simplify58.1

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

    3. Taylor expanded around -inf 2.2

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

    4. Applied simplify2.1

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

    if -6.601388653510457e+138 < t < 5.477946837407569e+42

    1. Initial program 37.4

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

    2. Applied simplify37.4

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

    3. Taylor expanded around inf 17.6

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

    4. Applied simplify13.1

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

    if 5.477946837407569e+42 < t

    1. Initial program 44.0

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

    2. Applied simplify44.0

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

    3. Taylor expanded around inf 4.1

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

    4. Applied simplify4.1

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

  4. Applied simplify9.0

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;t \le -6.601388653510457 \cdot 10^{+138}:\\ \;\;\;\;\frac{t}{\left(-t\right) - \frac{t}{2} \cdot \frac{2}{x}}\\ \mathbf{if}\;t \le 5.477946837407569 \cdot 10^{+42}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{(\left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) \cdot 2 + \left(\left(t \cdot t\right) \cdot \frac{4}{x}\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\frac{2}{x}}{\sqrt{2}} + \sqrt{2}}\\ \end{array}}\]

Runtime

Time bar (total: 2.0m)Debug logProfile

herbie shell --seed '#(1071948828 1180510430 2986424009 997076509 406109801 420189285)' +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)))))