Average Error: 12.7 → 8.9
Time: 50.8s
Precision: 64
Internal Precision: 128
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\frac{\sin th}{\frac{\sqrt{\left(\sin ky\right)^2 + \left(\sin kx\right)^2}^*}{\sin ky}}\]

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 12.7

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

  2. Initial simplification11.3

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

  4. Applied associate-/l*8.9

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

  5. Taylor expanded around -inf 12.7

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

  6. Simplified8.9

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

  7. Final simplification8.9

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

Runtime

Time bar (total: 50.8s)Debug logProfile

herbie shell --seed 2018348 +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)))