Average Error: 3.9 → 0.3

Time: 26.2s

Precision: 64

Internal Precision: 320

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

↓

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

Error

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

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Derivation

Initial program 3.9
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
Initial simplification2.4
\[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}\]
Using strategy rm
Applied associate-/l*0.2
\[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}{\sin ky}}}\]
Using strategy rm
Applied expm1-log1p-u0.3
\[\leadsto \color{blue}{(e^{\log_* (1 + \frac{\sin th}{\frac{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}{\sin ky}})} - 1)^*}\]
Final simplification0.3
\[\leadsto (e^{\log_* (1 + \frac{\sin th}{\frac{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}{\sin ky}})} - 1)^*\]

Runtime

	Baseline	Herbie	Oracle	Span	%
Regimes	0.3	0.3	0.0	0.2	0%

herbie shell --seed 2018290 +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)))