Average Error: 26.5 → 2.5
Time: 23.5s
Precision: 64
Internal Precision: 576
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{\cos \left(x \cdot 2\right)}{sin \cdot \left(cos \cdot x\right)} \cdot \frac{\frac{\frac{1}{x}}{cos}}{sin}\]

Error

Bits error versus x

Bits error versus cos

Bits error versus sin

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 26.5

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

  2. Initial simplification2.7

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

  4. Applied *-un-lft-identity2.7

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

  5. Applied times-frac2.4

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

  7. Applied associate-/r*2.5

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

  9. Applied associate-/r*2.5

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

  10. Final simplification2.5

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

Runtime

Time bar (total: 23.5s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes2.52.51.50.90%
herbie shell --seed 2018263 
(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))))