Average Error: 43.0 → 9.3
Time: 1.9m
Precision: 64
Internal Precision: 1344
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -8.426343753619193 \cdot 10^{+55}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{t \cdot \left(\left(-\sqrt{2}\right) - \frac{\frac{2}{x}}{\sqrt{2}}\right)}\\ \mathbf{elif}\;t \le -1.5688487797816682 \cdot 10^{-160}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\left(t \cdot t\right) \cdot \left(2 + \frac{4}{x}\right) + \frac{\ell \cdot 2}{\frac{x}{\ell}}}}\\ \mathbf{elif}\;t \le -8.120657760509964 \cdot 10^{-243}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{t \cdot \left(\left(-\sqrt{2}\right) - \frac{\frac{2}{x}}{\sqrt{2}}\right)}\\ \mathbf{elif}\;t \le 4.8469871076495545 \cdot 10^{-247}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\sqrt{\left(t \cdot t\right) \cdot \left(2 + \frac{4}{x}\right) + \frac{\ell \cdot 2}{\frac{x}{\ell}}}} \cdot \sqrt{\sqrt{\left(t \cdot t\right) \cdot \left(2 + \frac{4}{x}\right) + \frac{\ell \cdot 2}{\frac{x}{\ell}}}}}\\ \mathbf{elif}\;t \le 9.269948338767597 \cdot 10^{-174} \lor \neg \left(t \le 1.2204838220989629 \cdot 10^{+145}\right):\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\sqrt{2} + \frac{\frac{2}{x}}{\sqrt{2}}\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\left(t \cdot t\right) \cdot \left(2 + \frac{4}{x}\right) + \frac{\ell \cdot 2}{\frac{x}{\ell}}}}\\ \end{array}\]

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 < -8.426343753619193e+55 or -1.5688487797816682e-160 < t < -8.120657760509964e-243

    1. Initial program 48.8

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

    2. Initial simplification48.8

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

    3. Taylor expanded around -inf 9.3

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

    4. Simplified9.3

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

    if -8.426343753619193e+55 < t < -1.5688487797816682e-160 or 9.269948338767597e-174 < t < 1.2204838220989629e+145

    1. Initial program 27.0

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

    2. Initial simplification27.0

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

    3. Taylor expanded around -inf 10.8

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

    4. Simplified5.5

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

    if -8.120657760509964e-243 < t < 4.8469871076495545e-247

    1. Initial program 61.7

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

    2. Initial simplification61.7

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

    3. Taylor expanded around -inf 31.2

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

    4. Simplified30.8

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

    6. Applied add-sqr-sqrt30.8

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

    7. Applied sqrt-prod30.9

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

    if 4.8469871076495545e-247 < t < 9.269948338767597e-174 or 1.2204838220989629e+145 < t

    1. Initial program 60.0

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

    2. Initial simplification60.0

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

    3. Taylor expanded around inf 9.0

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

    4. Simplified9.1

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

  4. Final simplification9.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -8.426343753619193 \cdot 10^{+55}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{t \cdot \left(\left(-\sqrt{2}\right) - \frac{\frac{2}{x}}{\sqrt{2}}\right)}\\ \mathbf{elif}\;t \le -1.5688487797816682 \cdot 10^{-160}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\left(t \cdot t\right) \cdot \left(2 + \frac{4}{x}\right) + \frac{\ell \cdot 2}{\frac{x}{\ell}}}}\\ \mathbf{elif}\;t \le -8.120657760509964 \cdot 10^{-243}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{t \cdot \left(\left(-\sqrt{2}\right) - \frac{\frac{2}{x}}{\sqrt{2}}\right)}\\ \mathbf{elif}\;t \le 4.8469871076495545 \cdot 10^{-247}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\sqrt{\left(t \cdot t\right) \cdot \left(2 + \frac{4}{x}\right) + \frac{\ell \cdot 2}{\frac{x}{\ell}}}} \cdot \sqrt{\sqrt{\left(t \cdot t\right) \cdot \left(2 + \frac{4}{x}\right) + \frac{\ell \cdot 2}{\frac{x}{\ell}}}}}\\ \mathbf{elif}\;t \le 9.269948338767597 \cdot 10^{-174} \lor \neg \left(t \le 1.2204838220989629 \cdot 10^{+145}\right):\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\sqrt{2} + \frac{\frac{2}{x}}{\sqrt{2}}\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\left(t \cdot t\right) \cdot \left(2 + \frac{4}{x}\right) + \frac{\ell \cdot 2}{\frac{x}{\ell}}}}\\ \end{array}\]

Runtime

Time bar (total: 1.9m)Debug logProfile

herbie shell --seed 2018225 
(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)))))