Average Error: 27.4 → 3.1
Time: 4.5m
Precision: 64
Internal Precision: 128
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{\frac{\frac{1}{\sqrt[3]{\frac{sin \cdot \left(x \cdot cos\right)}{\cos \left(2 \cdot x\right)}} \cdot \sqrt[3]{\frac{sin \cdot \left(x \cdot cos\right)}{\cos \left(2 \cdot x\right)}}}}{\sqrt[3]{\frac{sin \cdot \left(x \cdot cos\right)}{\cos \left(2 \cdot x\right)}}}}{sin \cdot \left(x \cdot cos\right)}\]

Error

Bits error versus x

Bits error versus cos

Bits error versus sin

Derivation

  1. Initial program 27.4

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

  2. Simplified2.9

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

  4. Applied associate-/r*2.7

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

  6. Applied clear-num2.7

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

  8. Applied add-cube-cbrt3.1

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

  9. Applied associate-/r*3.1

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

  10. Final simplification3.1

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

Reproduce

herbie shell --seed 2019094 
(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))))