Average Error: 3.8 → 0.5
Time: 10.7s
Precision: 64
Internal precision: 384
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^2 + {\left(\sin ky\right)}^2}} \cdot \sin th\]
⬇
\[\begin{array}{l} \mathbf{if}\;{\left(\sin kx\right)}^2 + {\left(\sin ky\right)}^2 \le 9.8813129168249 \cdot 10^{-324}:\\ \;\;\;\;\frac{\sin ky}{\left(\frac{1}{12} \cdot \left({kx}^2 \cdot ky\right) + ky\right) - \frac{1}{6} \cdot {ky}^{3}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \left(\frac{1}{\sqrt{{\left(\sin kx\right)}^2 + {\left(\sin ky\right)}^2}} \cdot \sin th\right)\\ \end{array}\]

Error

Bits error versus kx

Bits error versus ky

Bits error versus th

Derivation

  1. Split input into 2 regimes.

  2. if (+ (sqr (sin kx)) (sqr (sin ky))) < 9.8813129168249e-324

    1. Initial program 62.9

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

    2. Applied taylor 0.0

      \[\leadsto \frac{\sin ky}{\left(\frac{1}{12} \cdot \left({kx}^2 \cdot ky\right) + ky\right) - \frac{1}{6} \cdot {ky}^{3}} \cdot \sin th\]

    3. Taylor expanded around 0 0.0

      \[\leadsto \frac{\sin ky}{\color{blue}{\left(\frac{1}{12} \cdot \left({kx}^2 \cdot ky\right) + ky\right) - \frac{1}{6} \cdot {ky}^{3}}} \cdot \sin th\]

    if 9.8813129168249e-324 < (+ (sqr (sin kx)) (sqr (sin ky)))

    1. Initial program 0.4

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

    3. Applied div-inv 0.5

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

    4. Applied associate-*l* 0.5

      \[\leadsto \color{blue}{\sin ky \cdot \left(\frac{1}{\sqrt{{\left(\sin kx\right)}^2 + {\left(\sin ky\right)}^2}} \cdot \sin th\right)}\]
  3. Recombined 2 regimes into one program.
  4. Removed slow pow expressions

Runtime

Time bar (total: 10.7s) Debug logProfile

Please include this information when filing a bug report:

herbie shell --seed '#(1064524629 4159152179 2999149171 575749698 4006532819 692958815)'
(FPCore (kx ky th)
  :name "Toniolo and Linder, Equation (3b), real"
  (* (/ (sin ky) (sqrt (+ (sqr (sin kx)) (sqr (sin ky))))) (sin th)))