Average Error: 27.9 → 2.9
Time: 30.6s
Precision: 64
Internal Precision: 576
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{{\left(\cos x\right)}^{6} - {\left(\sin x\right)}^{6}}{\left(\left(sin \cdot \left(cos \cdot x\right)\right) \cdot \left(sin \cdot \left(cos \cdot x\right)\right)\right) \cdot \left(\left(\left(\sin x \cdot \sin x\right) \cdot \left(\sin x \cdot \sin x\right) + \left(\cos x \cdot \cos x\right) \cdot \left(\sin x \cdot \sin x\right)\right) + \left(\cos x \cdot \cos x\right) \cdot \left(\cos x \cdot \cos x\right)\right)}\]

Error

Bits error versus x

Bits error versus cos

Bits error versus sin

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Initial simplification2.8

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

  4. Applied cos-22.9

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

  6. Applied flip3--2.9

    \[\leadsto \frac{\color{blue}{\frac{{\left(\cos x \cdot \cos x\right)}^{3} - {\left(\sin x \cdot \sin x\right)}^{3}}{\left(\cos x \cdot \cos x\right) \cdot \left(\cos x \cdot \cos x\right) + \left(\left(\sin x \cdot \sin x\right) \cdot \left(\sin x \cdot \sin x\right) + \left(\cos x \cdot \cos x\right) \cdot \left(\sin x \cdot \sin x\right)\right)}}}{\left(\left(x \cdot cos\right) \cdot sin\right) \cdot \left(\left(x \cdot cos\right) \cdot sin\right)}\]

  7. Applied associate-/l/2.9

    \[\leadsto \color{blue}{\frac{{\left(\cos x \cdot \cos x\right)}^{3} - {\left(\sin x \cdot \sin x\right)}^{3}}{\left(\left(\left(x \cdot cos\right) \cdot sin\right) \cdot \left(\left(x \cdot cos\right) \cdot sin\right)\right) \cdot \left(\left(\cos x \cdot \cos x\right) \cdot \left(\cos x \cdot \cos x\right) + \left(\left(\sin x \cdot \sin x\right) \cdot \left(\sin x \cdot \sin x\right) + \left(\cos x \cdot \cos x\right) \cdot \left(\sin x \cdot \sin x\right)\right)\right)}}\]

  8. Simplified2.9

    \[\leadsto \frac{\color{blue}{{\left(\cos x\right)}^{6} - {\left(\sin x\right)}^{6}}}{\left(\left(\left(x \cdot cos\right) \cdot sin\right) \cdot \left(\left(x \cdot cos\right) \cdot sin\right)\right) \cdot \left(\left(\cos x \cdot \cos x\right) \cdot \left(\cos x \cdot \cos x\right) + \left(\left(\sin x \cdot \sin x\right) \cdot \left(\sin x \cdot \sin x\right) + \left(\cos x \cdot \cos x\right) \cdot \left(\sin x \cdot \sin x\right)\right)\right)}\]

  9. Final simplification2.9

    \[\leadsto \frac{{\left(\cos x\right)}^{6} - {\left(\sin x\right)}^{6}}{\left(\left(sin \cdot \left(cos \cdot x\right)\right) \cdot \left(sin \cdot \left(cos \cdot x\right)\right)\right) \cdot \left(\left(\left(\sin x \cdot \sin x\right) \cdot \left(\sin x \cdot \sin x\right) + \left(\cos x \cdot \cos x\right) \cdot \left(\sin x \cdot \sin x\right)\right) + \left(\cos x \cdot \cos x\right) \cdot \left(\cos x \cdot \cos x\right)\right)}\]

Runtime

Time bar (total: 30.6s)Debug logProfile

herbie shell --seed 2018255 
(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))))