Average Error: 3.9 → 4.0

Time: 22.9s

Precision: 64

Internal Precision: 576

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

↓

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

Error

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

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Derivation

Initial program 3.9
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
Using strategy rm
Applied div-inv4.0
\[\leadsto \color{blue}{\left(\sin ky \cdot \frac{1}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}\right)} \cdot \sin th\]
Applied associate-*l*4.1
\[\leadsto \color{blue}{\sin ky \cdot \left(\frac{1}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\right)}\]
Using strategy rm
Applied associate-*r*4.0
\[\leadsto \color{blue}{\left(\sin ky \cdot \frac{1}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}\right) \cdot \sin th}\]
Final simplification4.0
\[\leadsto \left(\frac{1}{\sqrt{{\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}}} \cdot \sin ky\right) \cdot \sin th\]

Runtime

	Baseline	Herbie	Oracle	Span	%
Regimes	4.0	4.0	2.7	1.4	0%

herbie shell --seed 2018340 
(FPCore (kx ky th)
  :name "Toniolo and Linder, Equation (3b), real"
  (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))) (sin th)))