Average Error: 26.7 → 2.7
Time: 43.7s
Precision: 64
Internal Precision: 320
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{sin \cdot cos}}{x}}{\left(sin \cdot cos\right) \cdot x}\]

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 26.7

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

  2. Initial simplification3.0

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

  4. Applied div-inv3.0

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

  6. Applied pow13.0

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

  7. Applied pow13.0

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

  8. Applied pow-prod-down3.0

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

  9. Simplified2.8

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

  11. Applied associate-/r*2.6

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

  13. Applied associate-/r*2.7

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

  14. Final simplification2.7

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

Runtime

Time bar (total: 43.7s)Debug logProfile

herbie shell --seed 2018257 +o rules:numerics
(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))))