Average Error: 0.9 → 0.4
Time: 56.3s
Precision: 64
Internal Precision: 384
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\log_* (1 + (e^{\sqrt{1^2 + \left(\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^* \cdot \frac{\ell + \ell}{Om}\right)^2}^*} - 1)^*)}\right)}\]

Error

Bits error versus l

Bits error versus Om

Bits error versus kx

Bits error versus ky

Derivation

  1. Initial program 0.9

    \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
  2. Using strategy rm

  3. Applied add-exp-log0.9

    \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \color{blue}{e^{\log \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}}}\right)}\]

  4. Applied add-exp-log27.5

    \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\color{blue}{\left(e^{\log \left(\frac{2 \cdot \ell}{Om}\right)}\right)}}^{2} \cdot e^{\log \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}}\right)}\]

  5. Applied pow-exp27.5

    \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + \color{blue}{e^{\log \left(\frac{2 \cdot \ell}{Om}\right) \cdot 2}} \cdot e^{\log \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}}\right)}\]

  6. Applied prod-exp27.4

    \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + \color{blue}{e^{\log \left(\frac{2 \cdot \ell}{Om}\right) \cdot 2 + \log \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}}}\right)}\]
  7. Using strategy rm

  8. Applied add-sqr-sqrt27.4

    \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + e^{\log \left(\frac{2 \cdot \ell}{Om}\right) \cdot 2 + \log \color{blue}{\left(\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}} \cdot \sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}\right)}}}}\right)}\]

  9. Applied log-prod27.4

    \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + e^{\log \left(\frac{2 \cdot \ell}{Om}\right) \cdot 2 + \color{blue}{\left(\log \left(\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}\right) + \log \left(\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}\right)\right)}}}}\right)}\]

  10. Applied simplify27.4

    \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + e^{\log \left(\frac{2 \cdot \ell}{Om}\right) \cdot 2 + \left(\color{blue}{\log \left(\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*\right)} + \log \left(\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}\right)\right)}}}\right)}\]

  11. Applied simplify27.1

    \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + e^{\log \left(\frac{2 \cdot \ell}{Om}\right) \cdot 2 + \left(\log \left(\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*\right) + \color{blue}{\log \left(\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*\right)}\right)}}}\right)}\]
  12. Using strategy rm

  13. Applied log1p-expm1-u27.3

    \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\color{blue}{\log_* (1 + (e^{\sqrt{1 + e^{\log \left(\frac{2 \cdot \ell}{Om}\right) \cdot 2 + \left(\log \left(\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*\right) + \log \left(\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*\right)\right)}}} - 1)^*)}}\right)}\]

  14. Applied simplify0.4

    \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\log_* (1 + \color{blue}{(e^{\sqrt{1^2 + \left(\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^* \cdot \frac{\ell + \ell}{Om}\right)^2}^*} - 1)^*})}\right)}\]

Runtime

Time bar (total: 56.3s)Debug logProfile

herbie shell --seed '#(1070131407 1246090267 3027482374 2150728003 2026520792 2347815650)' +o rules:numerics
(FPCore (l Om kx ky)
  :name "Toniolo and Linder, Equation (3a)"
  (sqrt (* (/ 1 2) (+ 1 (/ 1 (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))))))