Average Error: 12.3 → 8.8
Time: 30.1s
Precision: 64
Internal Precision: 128
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[(e^{\log_* (1 + \sin th \cdot \log_* (1 + (e^{\frac{\sin ky}{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}} - 1)^*))} - 1)^*\]

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 12.3

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

  2. Simplified11.1

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

  4. Applied *-un-lft-identity11.1

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

  5. Applied times-frac8.8

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

  6. Simplified8.8

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

  8. Applied log1p-expm1-u8.8

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

  10. Applied expm1-log1p-u8.8

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

  11. Final simplification8.8

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

Reproduce

herbie shell --seed 2019026 +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)))