Average Error: 3.9 → 0.3
Time: 34.9s
Precision: 64
Internal Precision: 576
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\frac{\sin th}{\frac{\log_* (1 + (e^{\sqrt{\left(\sin ky\right)^2 + \left(\sin kx\right)^2}^*} - 1)^*)}{\sin ky}}\]

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 3.9

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

  2. Taylor expanded around -inf 3.9

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

  3. Simplified0.2

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

  5. Applied clear-num0.3

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

  7. Applied associate-*l/0.2

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

  8. Simplified0.2

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

  10. Applied log1p-expm1-u0.3

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

  11. Final simplification0.3

    \[\leadsto \frac{\sin th}{\frac{\log_* (1 + (e^{\sqrt{\left(\sin ky\right)^2 + \left(\sin kx\right)^2}^*} - 1)^*)}{\sin ky}}\]

Runtime

Time bar (total: 34.9s)Debug logProfile

herbie shell --seed 2018230 +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)))