Average Error: 3.9 → 4.3

Time: 25.5s

Precision: 64

Internal Precision: 320

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

↓

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

Error

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

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In
Out

Enter valid numbers for all inputs

Derivation

Initial program 3.9
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
Taylor expanded around inf 4.3
\[\leadsto \color{blue}{\left(\sqrt{\frac{1}{{\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}}} \cdot \sin ky\right)} \cdot \sin th\]
Final simplification4.3
\[\leadsto \left(\sin ky \cdot \sqrt{\frac{1}{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}\right) \cdot \sin th\]

Runtime

herbie shell --seed 2018248 
(FPCore (kx ky th)
  :name "Toniolo and Linder, Equation (3b), real"
  (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))) (sin th)))