Average Error: 12.5 → 8.8

Time: 47.5s

Precision: 64

Internal Precision: 128

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

↓

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

Error

The X axis uses an exponential scale — Bits error versus `kx`

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Derivation

Initial program 12.5
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
Initial simplification11.3
\[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}\]
Using strategy rm
Applied *-un-lft-identity11.3
\[\leadsto \frac{\sin th \cdot \sin ky}{\color{blue}{1 \cdot \sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}}\]
Applied times-frac8.8
\[\leadsto \color{blue}{\frac{\sin th}{1} \cdot \frac{\sin ky}{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}}\]
Simplified8.8
\[\leadsto \color{blue}{\sin th} \cdot \frac{\sin ky}{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}\]
Final simplification8.8
\[\leadsto \sin th \cdot \frac{\sin ky}{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}\]

Runtime

herbie shell --seed 2018349 +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)))