Average Error: 1.7 → 1.2
Time: 43.2s
Precision: 64
Internal Precision: 128
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2} \le 0.0:\\ \;\;\;\;\sqrt{\frac{1}{2} \cdot \left(\frac{1}{\sqrt{\left(4 \cdot \left(\ell \cdot \ell\right)\right) \cdot (\left(\frac{kx}{Om}\right) \cdot \left(\frac{kx}{Om}\right) + \left(\frac{ky}{Om} \cdot \frac{ky}{Om}\right))_* + 1}} + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(1 + \left(\sqrt[3]{\log \left(e^{\frac{1}{\sqrt{{\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right) + 1}}}\right)} \cdot \sqrt[3]{\log \left(e^{\frac{1}{\sqrt{{\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right) + 1}}}\right)}\right) \cdot \sqrt[3]{\log \left(e^{\frac{1}{\sqrt{{\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right) + 1}}}\right)}\right) \cdot \frac{1}{2}}\\ \end{array}\]

Error

Bits error versus l

Bits error versus Om

Bits error versus kx

Bits error versus ky

Derivation

  1. Split input into 2 regimes

  2. if (+ (pow (sin kx) 2) (pow (sin ky) 2)) < 0.0

    1. Initial program 18.1

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

    2. Taylor expanded around 0 27.6

      \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + \color{blue}{\left(4 \cdot \frac{{kx}^{2} \cdot {\ell}^{2}}{{Om}^{2}} + 4 \cdot \frac{{ky}^{2} \cdot {\ell}^{2}}{{Om}^{2}}\right)}}}\right)}\]

    3. Simplified8.9

      \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + \color{blue}{\left(\left(\ell \cdot \ell\right) \cdot 4\right) \cdot (\left(\frac{kx}{Om}\right) \cdot \left(\frac{kx}{Om}\right) + \left(\frac{ky}{Om} \cdot \frac{ky}{Om}\right))_*}}}\right)}\]

    if 0.0 < (+ (pow (sin kx) 2) (pow (sin ky) 2))

    1. Initial program 0.7

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

    3. Applied add-log-exp0.7

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

    5. Applied add-cube-cbrt0.7

      \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \color{blue}{\left(\sqrt[3]{\log \left(e^{\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}}\right)} \cdot \sqrt[3]{\log \left(e^{\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}}\right)}\right) \cdot \sqrt[3]{\log \left(e^{\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}}\right)}}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2} \le 0.0:\\ \;\;\;\;\sqrt{\frac{1}{2} \cdot \left(\frac{1}{\sqrt{\left(4 \cdot \left(\ell \cdot \ell\right)\right) \cdot (\left(\frac{kx}{Om}\right) \cdot \left(\frac{kx}{Om}\right) + \left(\frac{ky}{Om} \cdot \frac{ky}{Om}\right))_* + 1}} + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(1 + \left(\sqrt[3]{\log \left(e^{\frac{1}{\sqrt{{\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right) + 1}}}\right)} \cdot \sqrt[3]{\log \left(e^{\frac{1}{\sqrt{{\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right) + 1}}}\right)}\right) \cdot \sqrt[3]{\log \left(e^{\frac{1}{\sqrt{{\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right) + 1}}}\right)}\right) \cdot \frac{1}{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019030 +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))))))))))