Average Error: 12.5 → 12.5
Time: 40.1s
Precision: 64
Internal Precision: 128
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\frac{\sin ky}{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}} \cdot \sin th\]

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. Taylor expanded around -inf 12.8

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

  3. Simplified12.8

    \[\leadsto \color{blue}{\left(\sin ky \cdot \sqrt{\frac{1}{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}\right)} \cdot \sin th\]
  4. Using strategy rm

  5. Applied sqrt-div12.6

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

  6. Applied associate-*r/12.5

    \[\leadsto \color{blue}{\frac{\sin ky \cdot \sqrt{1}}{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}} \cdot \sin th\]

  7. Final simplification12.5

    \[\leadsto \frac{\sin ky}{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}} \cdot \sin th\]

Reproduce

herbie shell --seed 2019050 
(FPCore (kx ky th)
  :name "Toniolo and Linder, Equation (3b), real"
  (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))) (sin th)))