Average Error: 12.5 → 8.9
Time: 1.3m
Precision: 64
Internal Precision: 128
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\frac{\sin th}{\frac{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}{\sin ky}}\]

Error

Bits error versus kx

Bits error versus ky

Bits error versus th

Derivation

  1. Initial program 12.5

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

  2. Simplified8.9

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

  4. Applied *-un-lft-identity8.9

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

  5. Applied associate-/l*8.9

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

  6. Applied un-div-inv8.9

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

  7. Final simplification8.9

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

Reproduce

herbie shell --seed 2019072 +o rules:numerics
(FPCore (kx ky th)
  :name "Toniolo and Linder, Equation (3b), real"
  (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))) (sin th)))