Average Error: 42.3 → 9.3
Time: 58.1s
Precision: 64
Internal Precision: 1344
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -3.117428368383153 \cdot 10^{+112}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{t \cdot \left(\left(-\sqrt{2}\right) - \frac{\frac{2}{x}}{\sqrt{2}}\right)}\\ \mathbf{elif}\;t \le -6.590828423819892 \cdot 10^{-242}:\\ \;\;\;\;\frac{\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot t\right)}{\sqrt{\left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right) + \frac{2 \cdot \ell}{\frac{x}{\ell}}}}\\ \mathbf{elif}\;t \le -1.5989301694764613 \cdot 10^{-300}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{t \cdot \left(\left(-\sqrt{2}\right) - \frac{\frac{2}{x}}{\sqrt{2}}\right)}\\ \mathbf{elif}\;t \le 2.7483744020052034 \cdot 10^{-179}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right) + \frac{2}{\sqrt[3]{\frac{x}{\ell}} \cdot \sqrt[3]{\frac{x}{\ell}}} \cdot \frac{\ell}{\sqrt[3]{\frac{x}{\ell}}}}}\\ \mathbf{elif}\;t \le 1.4508073958694626 \cdot 10^{-156} \lor \neg \left(t \le 7.641252240928424 \cdot 10^{+125}\right):\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{t \cdot \left(\sqrt{2} + \frac{\frac{2}{x}}{\sqrt{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot t\right)}{\sqrt{\left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right) + \frac{2 \cdot \ell}{\frac{x}{\ell}}}}\\ \end{array}\]

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 < -3.117428368383153e+112 or -6.590828423819892e-242 < t < -1.5989301694764613e-300

    1. Initial program 54.7

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

    2. Initial simplification54.7

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

    3. Taylor expanded around -inf 9.2

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

    4. Simplified9.2

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

    if -3.117428368383153e+112 < t < -6.590828423819892e-242 or 1.4508073958694626e-156 < t < 7.641252240928424e+125

    1. Initial program 28.5

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

    2. Initial simplification28.5

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

    3. Taylor expanded around inf 12.4

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

    4. Simplified7.7

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

    5. Taylor expanded around inf 7.7

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

    7. Applied add-sqr-sqrt7.9

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

    8. Applied associate-*r*7.8

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

    if -1.5989301694764613e-300 < t < 2.7483744020052034e-179

    1. Initial program 61.5

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

    2. Initial simplification61.5

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

    3. Taylor expanded around inf 30.7

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

    4. Simplified28.8

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

    6. Applied add-cube-cbrt29.0

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

    7. Applied times-frac29.0

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

    if 2.7483744020052034e-179 < t < 1.4508073958694626e-156 or 7.641252240928424e+125 < t

    1. Initial program 55.0

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

    2. Initial simplification55.0

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

    3. Taylor expanded around inf 4.8

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

    4. Simplified4.8

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

  4. Final simplification9.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -3.117428368383153 \cdot 10^{+112}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{t \cdot \left(\left(-\sqrt{2}\right) - \frac{\frac{2}{x}}{\sqrt{2}}\right)}\\ \mathbf{elif}\;t \le -6.590828423819892 \cdot 10^{-242}:\\ \;\;\;\;\frac{\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot t\right)}{\sqrt{\left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right) + \frac{2 \cdot \ell}{\frac{x}{\ell}}}}\\ \mathbf{elif}\;t \le -1.5989301694764613 \cdot 10^{-300}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{t \cdot \left(\left(-\sqrt{2}\right) - \frac{\frac{2}{x}}{\sqrt{2}}\right)}\\ \mathbf{elif}\;t \le 2.7483744020052034 \cdot 10^{-179}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right) + \frac{2}{\sqrt[3]{\frac{x}{\ell}} \cdot \sqrt[3]{\frac{x}{\ell}}} \cdot \frac{\ell}{\sqrt[3]{\frac{x}{\ell}}}}}\\ \mathbf{elif}\;t \le 1.4508073958694626 \cdot 10^{-156} \lor \neg \left(t \le 7.641252240928424 \cdot 10^{+125}\right):\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{t \cdot \left(\sqrt{2} + \frac{\frac{2}{x}}{\sqrt{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot t\right)}{\sqrt{\left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right) + \frac{2 \cdot \ell}{\frac{x}{\ell}}}}\\ \end{array}\]

Runtime

Time bar (total: 58.1s)Debug logProfile

herbie shell --seed 2018234 
(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)))))