Average Error: 42.8 → 7.5
Time: 2.6m
Precision: 64
Internal Precision: 1344
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -3.135428033554305 \cdot 10^{+38}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\frac{t}{\sqrt{2} \cdot \left(x \cdot x\right)} \cdot \left(1 - 2\right) - t \cdot \left(\frac{\frac{2}{x}}{\sqrt{2}} + \sqrt{2}\right)}\\ \mathbf{if}\;t \le -7.96461310098498 \cdot 10^{-207}:\\ \;\;\;\;\frac{\left(t \cdot \sqrt{\sqrt{2}}\right) \cdot \sqrt{\sqrt{2}}}{\sqrt{\left(2 \cdot \ell\right) \cdot \frac{\ell}{x} + \left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)}}\\ \mathbf{if}\;t \le 2.986334750317526 \cdot 10^{-159}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\frac{\sqrt{\left(2 - \frac{4}{x}\right) \cdot \left(\left(2 + \frac{4}{x}\right) \cdot \left(\left(t \cdot x\right) \cdot t\right) + 2 \cdot \left(\ell \cdot \ell\right)\right)}}{\sqrt{x \cdot 2 - 4}}}\\ \mathbf{if}\;t \le 6.8042531379524426 \cdot 10^{+140}:\\ \;\;\;\;\frac{\left(t \cdot \sqrt{\sqrt{2}}\right) \cdot \sqrt{\sqrt{2}}}{\sqrt{\left(2 \cdot \ell\right) \cdot \frac{\ell}{x} + \left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\frac{\frac{t}{\sqrt{2}}}{x \cdot x} \cdot \left(2 - 1\right) + t \cdot \left(\frac{\frac{2}{x}}{\sqrt{2}} + \sqrt{2}\right)}\\ \end{array}\]

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 4 regimes

  2. if t < -3.135428033554305e+38

    1. Initial program 43.5

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

    2. Taylor expanded around -inf 4.1

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

    3. Applied simplify4.1

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

    if -3.135428033554305e+38 < t < -7.96461310098498e-207 or 2.986334750317526e-159 < t < 6.8042531379524426e+140

    1. Initial program 29.1

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

    2. Taylor expanded around inf 12.3

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

    3. Applied simplify6.6

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

    5. Applied add-sqr-sqrt6.8

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

    6. Applied associate-*r*6.7

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

    if -7.96461310098498e-207 < t < 2.986334750317526e-159

    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. Taylor expanded around inf 32.4

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

    3. Applied simplify31.1

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

    5. Applied flip-+31.1

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

    6. Applied associate-*l/31.1

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

    7. Applied associate-*r/32.4

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

    8. Applied frac-add32.8

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

    9. Applied sqrt-div27.5

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

    10. Applied simplify20.2

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

    11. Applied simplify20.2

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

    if 6.8042531379524426e+140 < t

    1. Initial program 58.5

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

    2. Taylor expanded around inf 2.1

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

    3. Applied simplify2.1

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

Runtime

Time bar (total: 2.6m)Debug logProfile

herbie shell --seed 2019053 
(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)))))