Average Error: 42.3 → 8.7
Time: 1.1m
Precision: 64
Internal Precision: 128
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -4.34365152350037 \cdot 10^{+49}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\frac{t}{x \cdot x} \cdot \frac{1}{\sqrt{2}} - \left(\frac{2}{\sqrt{2}} \cdot \left(\frac{t}{x} + \frac{t}{x \cdot x}\right) + \sqrt{2} \cdot t\right)}\\ \mathbf{elif}\;t \le -6.165739708625712 \cdot 10^{-209}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\left(\frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \left(2 \cdot \ell\right)\right) \cdot \frac{\ell}{\sqrt[3]{x}} + \left(t \cdot t\right) \cdot \left(2 + \frac{4}{x}\right)}}\\ \mathbf{elif}\;t \le -1.0388318465884581 \cdot 10^{-249}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\frac{t}{x \cdot x} \cdot \frac{1}{\sqrt{2}} - \left(\frac{2}{\sqrt{2}} \cdot \left(\frac{t}{x} + \frac{t}{x \cdot x}\right) + \sqrt{2} \cdot t\right)}\\ \mathbf{elif}\;t \le 1.8486538786678008 \cdot 10^{-196}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\frac{\sqrt{\left(\left(4 - \frac{4}{x} \cdot \left(2 - \frac{4}{x}\right)\right) \cdot \left(\ell \cdot \ell\right)\right) \cdot 2 + \left(x \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{\frac{64}{x \cdot x}}{x} + 8\right)}}{\sqrt{x \cdot \left(\left(4 - 2 \cdot \frac{4}{x}\right) + \frac{4}{x} \cdot \frac{4}{x}\right)}}}\\ \mathbf{elif}\;t \le 1.5573473699437125 \cdot 10^{-166}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\sqrt{2} \cdot t - \frac{t}{x \cdot x} \cdot \frac{1}{\sqrt{2}}\right) + \left(\frac{\frac{2}{x}}{x} + \frac{2}{x}\right) \cdot \frac{t}{\sqrt{2}}}\\ \mathbf{elif}\;t \le 1.4089337893440048 \cdot 10^{+125}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot t\right)\right)}{\sqrt{\left(t \cdot t\right) \cdot \left(2 + \frac{4}{x}\right) + \left(2 \cdot \ell\right) \cdot \frac{\ell}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\sqrt{2} \cdot t - \frac{t}{x \cdot x} \cdot \frac{1}{\sqrt{2}}\right) + \left(\frac{\frac{2}{x}}{x} + \frac{2}{x}\right) \cdot \frac{t}{\sqrt{2}}}\\ \end{array}\]

Error

Bits error versus x

Bits error versus l

Bits error versus t

Derivation

  1. Split input into 5 regimes

  2. if t < -4.34365152350037e+49 or -6.165739708625712e-209 < t < -1.0388318465884581e-249

    1. Initial program 45.5

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

    2. Taylor expanded around -inf 6.7

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

    3. Simplified6.7

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

    if -4.34365152350037e+49 < t < -6.165739708625712e-209

    1. Initial program 32.8

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

    2. Taylor expanded around -inf 14.0

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

    3. Simplified9.7

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

    5. Applied add-cube-cbrt9.8

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

    6. Applied *-un-lft-identity9.8

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\left(t \cdot t\right) \cdot \left(\frac{4}{x} + 2\right) + \left(\ell \cdot 2\right) \cdot \frac{\color{blue}{1 \cdot \ell}}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}}\]

    7. Applied times-frac9.8

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\left(t \cdot t\right) \cdot \left(\frac{4}{x} + 2\right) + \left(\ell \cdot 2\right) \cdot \color{blue}{\left(\frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\ell}{\sqrt[3]{x}}\right)}}}\]

    8. Applied associate-*r*9.8

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\left(t \cdot t\right) \cdot \left(\frac{4}{x} + 2\right) + \color{blue}{\left(\left(\ell \cdot 2\right) \cdot \frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right) \cdot \frac{\ell}{\sqrt[3]{x}}}}}\]

    if -1.0388318465884581e-249 < t < 1.8486538786678008e-196

    1. Initial program 61.1

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

    2. Taylor expanded around -inf 30.5

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

    3. Simplified29.5

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

    5. Applied associate-*r/30.5

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

    6. Applied flip3-+30.5

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

    7. Applied associate-*r/30.5

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

    8. Applied frac-add30.7

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

    9. Applied sqrt-div25.2

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

    10. Simplified25.2

      \[\leadsto \frac{\sqrt{2} \cdot t}{\frac{\color{blue}{\sqrt{\left(\left(4 - \frac{4}{x} \cdot \left(2 - \frac{4}{x}\right)\right) \cdot \left(\ell \cdot \ell\right)\right) \cdot 2 + \left(x \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{\frac{64}{x \cdot x}}{x} + 8\right)}}}{\sqrt{\left(\frac{4}{x} \cdot \frac{4}{x} + \left(2 \cdot 2 - \frac{4}{x} \cdot 2\right)\right) \cdot x}}}\]

    if 1.8486538786678008e-196 < t < 1.5573473699437125e-166 or 1.4089337893440048e+125 < t

    1. Initial program 56.0

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

    2. Taylor expanded around inf 5.2

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

    3. Simplified5.2

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

    if 1.5573473699437125e-166 < t < 1.4089337893440048e+125

    1. Initial program 26.1

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

    2. Taylor expanded around -inf 11.3

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

    3. Simplified6.1

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

    5. Applied add-cube-cbrt6.1

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

    6. Applied associate-*l*6.1

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

    8. Applied add-cube-cbrt6.1

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

    9. Applied associate-*l*6.1

      \[\leadsto \frac{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot t\right)\right)}}{\sqrt{\left(t \cdot t\right) \cdot \left(\frac{4}{x} + 2\right) + \left(\ell \cdot 2\right) \cdot \frac{\ell}{x}}}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification8.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -4.34365152350037 \cdot 10^{+49}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\frac{t}{x \cdot x} \cdot \frac{1}{\sqrt{2}} - \left(\frac{2}{\sqrt{2}} \cdot \left(\frac{t}{x} + \frac{t}{x \cdot x}\right) + \sqrt{2} \cdot t\right)}\\ \mathbf{elif}\;t \le -6.165739708625712 \cdot 10^{-209}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\left(\frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \left(2 \cdot \ell\right)\right) \cdot \frac{\ell}{\sqrt[3]{x}} + \left(t \cdot t\right) \cdot \left(2 + \frac{4}{x}\right)}}\\ \mathbf{elif}\;t \le -1.0388318465884581 \cdot 10^{-249}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\frac{t}{x \cdot x} \cdot \frac{1}{\sqrt{2}} - \left(\frac{2}{\sqrt{2}} \cdot \left(\frac{t}{x} + \frac{t}{x \cdot x}\right) + \sqrt{2} \cdot t\right)}\\ \mathbf{elif}\;t \le 1.8486538786678008 \cdot 10^{-196}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\frac{\sqrt{\left(\left(4 - \frac{4}{x} \cdot \left(2 - \frac{4}{x}\right)\right) \cdot \left(\ell \cdot \ell\right)\right) \cdot 2 + \left(x \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{\frac{64}{x \cdot x}}{x} + 8\right)}}{\sqrt{x \cdot \left(\left(4 - 2 \cdot \frac{4}{x}\right) + \frac{4}{x} \cdot \frac{4}{x}\right)}}}\\ \mathbf{elif}\;t \le 1.5573473699437125 \cdot 10^{-166}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\sqrt{2} \cdot t - \frac{t}{x \cdot x} \cdot \frac{1}{\sqrt{2}}\right) + \left(\frac{\frac{2}{x}}{x} + \frac{2}{x}\right) \cdot \frac{t}{\sqrt{2}}}\\ \mathbf{elif}\;t \le 1.4089337893440048 \cdot 10^{+125}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot t\right)\right)}{\sqrt{\left(t \cdot t\right) \cdot \left(2 + \frac{4}{x}\right) + \left(2 \cdot \ell\right) \cdot \frac{\ell}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\sqrt{2} \cdot t - \frac{t}{x \cdot x} \cdot \frac{1}{\sqrt{2}}\right) + \left(\frac{\frac{2}{x}}{x} + \frac{2}{x}\right) \cdot \frac{t}{\sqrt{2}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019090 
(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)))))