Average Error: 27.9 → 2.7
Time: 44.5s
Precision: 64
Internal Precision: 128
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{\frac{1}{sin} \cdot \left(\frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{cos} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{x}\right)}{sin \cdot \left(x \cdot cos\right)}\]

Error

Bits error versus x

Bits error versus cos

Bits error versus sin

Derivation

  1. Initial program 27.9

    \[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]

  2. Simplified2.7

    \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{\left(sin \cdot \left(x \cdot cos\right)\right) \cdot \left(sin \cdot \left(x \cdot cos\right)\right)}}\]
  3. Using strategy rm

  4. Applied associate-/r*2.4

    \[\leadsto \color{blue}{\frac{\frac{\cos \left(2 \cdot x\right)}{sin \cdot \left(x \cdot cos\right)}}{sin \cdot \left(x \cdot cos\right)}}\]
  5. Using strategy rm

  6. Applied *-un-lft-identity2.4

    \[\leadsto \frac{\frac{\color{blue}{1 \cdot \cos \left(2 \cdot x\right)}}{sin \cdot \left(x \cdot cos\right)}}{sin \cdot \left(x \cdot cos\right)}\]

  7. Applied times-frac2.6

    \[\leadsto \frac{\color{blue}{\frac{1}{sin} \cdot \frac{\cos \left(2 \cdot x\right)}{x \cdot cos}}}{sin \cdot \left(x \cdot cos\right)}\]
  8. Using strategy rm

  9. Applied add-cube-cbrt2.8

    \[\leadsto \frac{\frac{1}{sin} \cdot \frac{\color{blue}{\left(\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}\right) \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}}{x \cdot cos}}{sin \cdot \left(x \cdot cos\right)}\]

  10. Applied times-frac2.7

    \[\leadsto \frac{\frac{1}{sin} \cdot \color{blue}{\left(\frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{x} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{cos}\right)}}{sin \cdot \left(x \cdot cos\right)}\]

  11. Final simplification2.7

    \[\leadsto \frac{\frac{1}{sin} \cdot \left(\frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{cos} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{x}\right)}{sin \cdot \left(x \cdot cos\right)}\]

Reproduce

herbie shell --seed 2019089 
(FPCore (x cos sin)
  :name "cos(2*x)/(cos^2(x)*sin^2(x))"
  (/ (cos (* 2 x)) (* (pow cos 2) (* (* x (pow sin 2)) x))))