Average Error: 4.0 → 0.5
Time: 38.9s
Precision: 64
Internal Precision: 576
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\frac{\sin ky}{\sqrt{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}} \cdot \frac{\sin th}{\sqrt{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}}\]

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 4.0

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

  2. Initial simplification2.4

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

  4. Applied add-sqr-sqrt2.7

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

  5. Applied times-frac0.5

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

  6. Final simplification0.5

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

Runtime

Time bar (total: 38.9s)Debug logProfile

herbie shell --seed 2018227 +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)))