Average Error: 13.1 → 9.4
Time: 52.0s
Precision: 64
Internal Precision: 128
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\frac{\sin th}{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*} \cdot \sin ky\]

Error

Bits error versus kx

Bits error versus ky

Bits error versus th

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 13.1

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

  2. Initial simplification11.7

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

  4. Applied associate-/l*9.4

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

  6. Applied div-inv9.4

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

  7. Applied associate-/r*9.5

    \[\leadsto \color{blue}{\frac{\frac{\sin th}{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}}{\frac{1}{\sin ky}}}\]
  8. Using strategy rm

  9. Applied associate-/r/9.4

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

  10. Final simplification9.4

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

Runtime

Time bar (total: 52.0s)Debug logProfile

herbie shell --seed 2018277 +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)))