Average Error: 42.2 → 9.3
Time: 2.7m
Precision: 64
Internal Precision: 1344
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -7.154720090399836 \cdot 10^{+99}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\frac{\frac{t}{\sqrt{2}}}{x \cdot x} \cdot \left(1 - 2\right) - t \cdot \left(\sqrt{2} + \frac{\frac{2}{x}}{\sqrt{2}}\right)}\\ \mathbf{if}\;t \le -4.603222059734245 \cdot 10^{-150}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left|\sqrt[3]{\left(\ell \cdot 2\right) \cdot \frac{\ell}{x} + \left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)}\right| \cdot \sqrt{\sqrt[3]{\left(\frac{4}{x} + 2\right) \cdot \left(t \cdot t\right) + \frac{2 \cdot \ell}{\frac{x}{\ell}}}}}\\ \mathbf{if}\;t \le -4.667265153057396 \cdot 10^{-240}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\frac{\frac{t}{\sqrt{2}}}{x \cdot x} \cdot \left(1 - 2\right) - t \cdot \left(\sqrt{2} + \frac{\frac{2}{x}}{\sqrt{2}}\right)}\\ \mathbf{if}\;t \le 1.7818222536172002 \cdot 10^{+127}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left|\sqrt[3]{\left(\ell \cdot 2\right) \cdot \frac{\ell}{x} + \left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)}\right| \cdot \sqrt{\sqrt[3]{\left(\frac{4}{x} + 2\right) \cdot \left(t \cdot t\right) + \frac{2 \cdot \ell}{\frac{x}{\ell}}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \left(\sqrt{2} + \frac{\frac{2}{x}}{\sqrt{2}}\right) + \frac{t}{\left(x \cdot x\right) \cdot \sqrt{2}} \cdot \left(2 - 1\right)}\\ \end{array}\]

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 3 regimes

  2. if t < -7.154720090399836e+99 or -4.603222059734245e-150 < t < -4.667265153057396e-240

    1. Initial program 52.7

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

    2. Taylor expanded around -inf 9.1

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

    3. Applied simplify9.2

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

    if -7.154720090399836e+99 < t < -4.603222059734245e-150 or -4.667265153057396e-240 < t < 1.7818222536172002e+127

    1. Initial program 33.3

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

    2. Taylor expanded around inf 15.3

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

    3. Applied simplify11.1

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

    5. Applied add-cube-cbrt11.4

      \[\leadsto \frac{t \cdot \sqrt{2}}{\sqrt{\color{blue}{\left(\sqrt[3]{\left(\frac{4}{x} + 2\right) \cdot \left(t \cdot t\right) + \frac{2 \cdot \ell}{\frac{x}{\ell}}} \cdot \sqrt[3]{\left(\frac{4}{x} + 2\right) \cdot \left(t \cdot t\right) + \frac{2 \cdot \ell}{\frac{x}{\ell}}}\right) \cdot \sqrt[3]{\left(\frac{4}{x} + 2\right) \cdot \left(t \cdot t\right) + \frac{2 \cdot \ell}{\frac{x}{\ell}}}}}}\]

    6. Applied sqrt-prod11.4

      \[\leadsto \frac{t \cdot \sqrt{2}}{\color{blue}{\sqrt{\sqrt[3]{\left(\frac{4}{x} + 2\right) \cdot \left(t \cdot t\right) + \frac{2 \cdot \ell}{\frac{x}{\ell}}} \cdot \sqrt[3]{\left(\frac{4}{x} + 2\right) \cdot \left(t \cdot t\right) + \frac{2 \cdot \ell}{\frac{x}{\ell}}}} \cdot \sqrt{\sqrt[3]{\left(\frac{4}{x} + 2\right) \cdot \left(t \cdot t\right) + \frac{2 \cdot \ell}{\frac{x}{\ell}}}}}}\]

    7. Applied simplify11.4

      \[\leadsto \frac{t \cdot \sqrt{2}}{\color{blue}{\left|\sqrt[3]{\left(\ell \cdot 2\right) \cdot \frac{\ell}{x} + \left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)}\right|} \cdot \sqrt{\sqrt[3]{\left(\frac{4}{x} + 2\right) \cdot \left(t \cdot t\right) + \frac{2 \cdot \ell}{\frac{x}{\ell}}}}}\]

    if 1.7818222536172002e+127 < t

    1. Initial program 55.1

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

    2. Taylor expanded around inf 2.7

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

    3. Applied simplify2.7

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

Runtime

Time bar (total: 2.7m)Debug logProfile

herbie shell --seed 2018199 
(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)))))