Average Error: 3.8 → 0.2
Time: 27.9s
Precision: 64
Internal Precision: 320
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\sin th \cdot \log_* (1 + (e^{\frac{\sin ky}{\sqrt{\left(\sin ky\right)^2 + \left(\sin kx\right)^2}^*}} - 1)^*)\]

Error

Bits error versus kx

Bits error versus ky

Bits error versus th

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 3.8

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

  2. Taylor expanded around inf 3.8

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

  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 log1p-expm1-u0.2

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

  6. Final simplification0.2

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

Runtime

Time bar (total: 27.9s)Debug logProfile

herbie shell --seed 2018254 +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)))