Average Error: 27.8 → 3.0
Time: 24.1s
Precision: 64
Internal Precision: 128
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{\cos \left(2 \cdot x\right)}{\left(\left(x \cdot sin\right) \cdot cos\right) \cdot \left(\left(x \cdot sin\right) \cdot cos\right)}\]

Error

Bits error versus x

Bits error versus cos

Bits error versus sin

Derivation

  1. Initial program 27.8

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

  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. Taylor expanded around inf 31.1

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

  4. Simplified3.0

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

  6. Applied associate-/r*2.7

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

  7. Taylor expanded around inf 31.2

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

  8. Simplified3.0

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

  9. Final simplification3.0

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

Reproduce

herbie shell --seed 2019093 
(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))))