Average Error: 12.7 → 9.2

Time: 41.3s

Precision: 64

Internal Precision: 128

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

↓

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

Error

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

Try it out

In
Out

Enter valid numbers for all inputs

Derivation

Initial program 12.7
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
Initial simplification11.7
\[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}\]
Using strategy rm
Applied associate-/l*9.0
\[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}{\sin ky}}}\]
Using strategy rm
Applied clear-num9.2
\[\leadsto \color{blue}{\frac{1}{\frac{\frac{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}{\sin ky}}{\sin th}}}\]
Final simplification9.2
\[\leadsto \frac{1}{\frac{\frac{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}{\sin ky}}{\sin th}}\]

Runtime

	Baseline	Herbie	Oracle	Span	%
Regimes	9.2	9.2	8.9	0.3	0%

herbie shell --seed 2018353 +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)))