Average Error: 32.9 → 8.9
Time: 1.6m
Precision: 64
Internal Precision: 384
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -2.6159991165134946 \cdot 10^{-67}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\frac{k}{t} \cdot k + t \cdot 2\right) \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \sin k\right)}{\cos k}}\\ \mathbf{if}\;t \le 2.5007688837872216 \cdot 10^{-106}:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{{k}^{2} \cdot \sin k}{\ell} + 2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \sin k\right)}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\frac{k}{t} \cdot k + t \cdot 2\right) \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \sin k\right)}{\cos k}}\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Split input into 2 regimes

  2. if t < -2.6159991165134946e-67 or 2.5007688837872216e-106 < t

    1. Initial program 23.5

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
    2. Using strategy rm

    3. Applied add-cbrt-cube27.2

      \[\leadsto \frac{2}{\color{blue}{\sqrt[3]{\left(\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right) \cdot \left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}}\]

    4. Applied simplify20.8

      \[\leadsto \frac{2}{\sqrt[3]{\color{blue}{{\left(\left(\left(\frac{t}{\ell} \cdot \sin k\right) \cdot \left(\frac{t}{\ell} \cdot t\right)\right) \cdot \left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_* \cdot \tan k\right)\right)}^{3}}}}\]
    5. Using strategy rm

    6. Applied tan-quot20.8

      \[\leadsto \frac{2}{\sqrt[3]{{\left(\left(\left(\frac{t}{\ell} \cdot \sin k\right) \cdot \left(\frac{t}{\ell} \cdot t\right)\right) \cdot \left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_* \cdot \color{blue}{\frac{\sin k}{\cos k}}\right)\right)}^{3}}}\]

    7. Applied associate-*r/20.8

      \[\leadsto \frac{2}{\sqrt[3]{{\left(\left(\left(\frac{t}{\ell} \cdot \sin k\right) \cdot \left(\frac{t}{\ell} \cdot t\right)\right) \cdot \color{blue}{\frac{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_* \cdot \sin k}{\cos k}}\right)}^{3}}}\]

    8. Applied associate-*r/20.8

      \[\leadsto \frac{2}{\sqrt[3]{{\color{blue}{\left(\frac{\left(\left(\frac{t}{\ell} \cdot \sin k\right) \cdot \left(\frac{t}{\ell} \cdot t\right)\right) \cdot \left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_* \cdot \sin k\right)}{\cos k}\right)}}^{3}}}\]

    9. Applied cube-div20.8

      \[\leadsto \frac{2}{\sqrt[3]{\color{blue}{\frac{{\left(\left(\left(\frac{t}{\ell} \cdot \sin k\right) \cdot \left(\frac{t}{\ell} \cdot t\right)\right) \cdot \left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_* \cdot \sin k\right)\right)}^{3}}{{\left(\cos k\right)}^{3}}}}}\]

    10. Applied cbrt-div20.8

      \[\leadsto \frac{2}{\color{blue}{\frac{\sqrt[3]{{\left(\left(\left(\frac{t}{\ell} \cdot \sin k\right) \cdot \left(\frac{t}{\ell} \cdot t\right)\right) \cdot \left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_* \cdot \sin k\right)\right)}^{3}}}{\sqrt[3]{{\left(\cos k\right)}^{3}}}}}\]

    11. Applied simplify8.0

      \[\leadsto \frac{2}{\frac{\color{blue}{\left(t \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*\right) \cdot \left(\left(\frac{t}{\ell} \cdot \sin k\right) \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)}}{\sqrt[3]{{\left(\cos k\right)}^{3}}}}\]

    12. Applied simplify7.9

      \[\leadsto \frac{2}{\frac{\left(t \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*\right) \cdot \left(\left(\frac{t}{\ell} \cdot \sin k\right) \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)}{\color{blue}{\cos k}}}\]
    13. Using strategy rm

    14. Applied associate-*r*5.0

      \[\leadsto \frac{2}{\frac{\color{blue}{\left(\left(t \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*\right) \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \sin k\right)}}{\cos k}}\]
    15. Using strategy rm

    16. Applied fma-udef5.0

      \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \color{blue}{\left(\frac{k}{t} \cdot \frac{k}{t} + 2\right)}\right) \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \sin k\right)}{\cos k}}\]

    17. Applied distribute-lft-in5.0

      \[\leadsto \frac{2}{\frac{\left(\color{blue}{\left(t \cdot \left(\frac{k}{t} \cdot \frac{k}{t}\right) + t \cdot 2\right)} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \sin k\right)}{\cos k}}\]

    18. Applied simplify4.7

      \[\leadsto \frac{2}{\frac{\left(\left(\color{blue}{\frac{k}{t} \cdot k} + t \cdot 2\right) \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \sin k\right)}{\cos k}}\]

    if -2.6159991165134946e-67 < t < 2.5007688837872216e-106

    1. Initial program 59.6

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
    2. Using strategy rm

    3. Applied add-cbrt-cube60.0

      \[\leadsto \frac{2}{\color{blue}{\sqrt[3]{\left(\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right) \cdot \left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}}\]

    4. Applied simplify44.9

      \[\leadsto \frac{2}{\sqrt[3]{\color{blue}{{\left(\left(\left(\frac{t}{\ell} \cdot \sin k\right) \cdot \left(\frac{t}{\ell} \cdot t\right)\right) \cdot \left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_* \cdot \tan k\right)\right)}^{3}}}}\]
    5. Using strategy rm

    6. Applied tan-quot44.9

      \[\leadsto \frac{2}{\sqrt[3]{{\left(\left(\left(\frac{t}{\ell} \cdot \sin k\right) \cdot \left(\frac{t}{\ell} \cdot t\right)\right) \cdot \left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_* \cdot \color{blue}{\frac{\sin k}{\cos k}}\right)\right)}^{3}}}\]

    7. Applied associate-*r/44.9

      \[\leadsto \frac{2}{\sqrt[3]{{\left(\left(\left(\frac{t}{\ell} \cdot \sin k\right) \cdot \left(\frac{t}{\ell} \cdot t\right)\right) \cdot \color{blue}{\frac{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_* \cdot \sin k}{\cos k}}\right)}^{3}}}\]

    8. Applied associate-*r/44.9

      \[\leadsto \frac{2}{\sqrt[3]{{\color{blue}{\left(\frac{\left(\left(\frac{t}{\ell} \cdot \sin k\right) \cdot \left(\frac{t}{\ell} \cdot t\right)\right) \cdot \left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_* \cdot \sin k\right)}{\cos k}\right)}}^{3}}}\]

    9. Applied cube-div44.9

      \[\leadsto \frac{2}{\sqrt[3]{\color{blue}{\frac{{\left(\left(\left(\frac{t}{\ell} \cdot \sin k\right) \cdot \left(\frac{t}{\ell} \cdot t\right)\right) \cdot \left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_* \cdot \sin k\right)\right)}^{3}}{{\left(\cos k\right)}^{3}}}}}\]

    10. Applied cbrt-div44.9

      \[\leadsto \frac{2}{\color{blue}{\frac{\sqrt[3]{{\left(\left(\left(\frac{t}{\ell} \cdot \sin k\right) \cdot \left(\frac{t}{\ell} \cdot t\right)\right) \cdot \left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_* \cdot \sin k\right)\right)}^{3}}}{\sqrt[3]{{\left(\cos k\right)}^{3}}}}}\]

    11. Applied simplify35.4

      \[\leadsto \frac{2}{\frac{\color{blue}{\left(t \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*\right) \cdot \left(\left(\frac{t}{\ell} \cdot \sin k\right) \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)}}{\sqrt[3]{{\left(\cos k\right)}^{3}}}}\]

    12. Applied simplify35.4

      \[\leadsto \frac{2}{\frac{\left(t \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*\right) \cdot \left(\left(\frac{t}{\ell} \cdot \sin k\right) \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)}{\color{blue}{\cos k}}}\]
    13. Using strategy rm

    14. Applied associate-*r*32.9

      \[\leadsto \frac{2}{\frac{\color{blue}{\left(\left(t \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*\right) \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \sin k\right)}}{\cos k}}\]

    15. Taylor expanded around inf 20.8

      \[\leadsto \frac{2}{\frac{\color{blue}{\left(\frac{{k}^{2} \cdot \sin k}{\ell} + 2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell}\right)} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)}{\cos k}}\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 1.6m)Debug logProfile

herbie shell --seed '#(1070100504 930361288 1279167582 284574201 1450237281 2578255382)' +o rules:numerics
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  (/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))))