Average Error: 3.8 → 4.0
Time: 1.5m
Precision: 64
Internal Precision: 384
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\frac{\sin ky}{\sqrt{\left(\sin kx \cdot \sqrt[3]{\sin kx}\right) \cdot {\left(\sqrt[3]{\sin kx}\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]

Error

Bits error versus kx

Bits error versus ky

Bits error versus th

Derivation

  1. Initial program 3.8

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

  3. Applied add-cube-cbrt4.2

    \[\leadsto \frac{\sin ky}{\sqrt{{\color{blue}{\left(\left(\sqrt[3]{\sin kx} \cdot \sqrt[3]{\sin kx}\right) \cdot \sqrt[3]{\sin kx}\right)}}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]

  4. Applied unpow-prod-down4.2

    \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\left(\sqrt[3]{\sin kx} \cdot \sqrt[3]{\sin kx}\right)}^{2} \cdot {\left(\sqrt[3]{\sin kx}\right)}^{2}} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]

  5. Applied simplify4.0

    \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\sin kx \cdot \sqrt[3]{\sin kx}\right)} \cdot {\left(\sqrt[3]{\sin kx}\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
  6. Removed slow pow expressions.

Runtime

Time bar (total: 1.5m)Debug logProfile

herbie shell --seed '#(1063027428 1192549564 1443466578 604016274 3637110559 1698629644)' 
(FPCore (kx ky th)
  :name "Toniolo and Linder, Equation (3b), real"
  (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))) (sin th)))