Average Error: 42.2 → 8.9
Time: 2.5m
Precision: 64
Internal Precision: 1344
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -2.5592594841895054 \cdot 10^{+128}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(1 - 2\right) \cdot \frac{\frac{t}{\sqrt{2}}}{x \cdot x} - \left(\frac{\frac{2}{x}}{\sqrt{2}} + \sqrt{2}\right) \cdot t}\\ \mathbf{if}\;t \le -9.864011027762625 \cdot 10^{-157}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{\ell \cdot 2}{\frac{x}{\ell}} + \left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)}}\\ \mathbf{if}\;t \le -9.608770765623839 \cdot 10^{-244}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(1 - 2\right) \cdot \frac{\frac{t}{\sqrt{2}}}{x \cdot x} - \left(\frac{\frac{2}{x}}{\sqrt{2}} + \sqrt{2}\right) \cdot t}\\ \mathbf{if}\;t \le 2.904625615805768 \cdot 10^{-261} \lor \neg \left(t \le 4.6083392674133244 \cdot 10^{-161} \lor \neg \left(t \le 5.237390130902631 \cdot 10^{+98}\right)\right):\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{\ell \cdot 2}{\frac{x}{\ell}} + \left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\frac{t}{\left(x \cdot x\right) \cdot \sqrt{2}} \cdot \left(2 - 1\right) + \left(\frac{\frac{2}{x}}{\sqrt{2}} + \sqrt{2}\right) \cdot t}\\ \end{array}\]

Error

Bits error versus x

Bits error versus l

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 4 regimes

  2. if t < -2.5592594841895054e+128

    1. Initial program 55.8

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

    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 simplify2.6

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

    if -2.5592594841895054e+128 < t < -9.864011027762625e-157 or -9.608770765623839e-244 < t < 2.904625615805768e-261 or 4.6083392674133244e-161 < t < 5.237390130902631e+98

    1. Initial program 30.2

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

    2. Taylor expanded around inf 12.0

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

    3. Applied simplify7.8

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

    if -9.864011027762625e-157 < t < -9.608770765623839e-244

    1. Initial program 60.4

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

    2. Taylor expanded around -inf 31.5

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

    3. Applied simplify31.5

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

    if 2.904625615805768e-261 < t < 4.6083392674133244e-161 or 5.237390130902631e+98 < t

    1. Initial program 52.5

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

    2. Taylor expanded around inf 10.3

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

    3. Applied simplify10.3

      \[\leadsto \color{blue}{\frac{t \cdot \sqrt{2}}{t \cdot \left(\sqrt{2} + \frac{\frac{2}{x}}{\sqrt{2}}\right) + \frac{t}{\left(x \cdot x\right) \cdot \sqrt{2}} \cdot \left(2 - 1\right)}}\]
  3. Recombined 4 regimes into one program.

  4. Applied simplify8.9

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;t \le -2.5592594841895054 \cdot 10^{+128}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(1 - 2\right) \cdot \frac{\frac{t}{\sqrt{2}}}{x \cdot x} - \left(\frac{\frac{2}{x}}{\sqrt{2}} + \sqrt{2}\right) \cdot t}\\ \mathbf{if}\;t \le -9.864011027762625 \cdot 10^{-157}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{\ell \cdot 2}{\frac{x}{\ell}} + \left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)}}\\ \mathbf{if}\;t \le -9.608770765623839 \cdot 10^{-244}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(1 - 2\right) \cdot \frac{\frac{t}{\sqrt{2}}}{x \cdot x} - \left(\frac{\frac{2}{x}}{\sqrt{2}} + \sqrt{2}\right) \cdot t}\\ \mathbf{if}\;t \le 2.904625615805768 \cdot 10^{-261} \lor \neg \left(t \le 4.6083392674133244 \cdot 10^{-161} \lor \neg \left(t \le 5.237390130902631 \cdot 10^{+98}\right)\right):\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{\ell \cdot 2}{\frac{x}{\ell}} + \left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\frac{t}{\left(x \cdot x\right) \cdot \sqrt{2}} \cdot \left(2 - 1\right) + \left(\frac{\frac{2}{x}}{\sqrt{2}} + \sqrt{2}\right) \cdot t}\\ \end{array}}\]

Runtime

Time bar (total: 2.5m)Debug logProfile

herbie shell --seed 2018166 
(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)))))