Average Error: 27.2 → 2.8
Time: 30.3s
Precision: 64
Internal Precision: 576
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{\frac{\cos \left(x \cdot 2\right)}{cos \cdot x}}{sin} \cdot \frac{1}{\left(cos \cdot x\right) \cdot sin}\]

Error

Bits error versus x

Bits error versus cos

Bits error versus sin

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Results

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Derivation

  1. Initial program 27.2

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

  2. Initial simplification2.9

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

  4. Applied *-un-lft-identity2.9

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

  5. Applied times-frac2.7

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

  7. Applied associate-/r*2.8

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

  8. Final simplification2.8

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

Runtime

Time bar (total: 30.3s)Debug logProfile

herbie shell --seed 2018249 
(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))))