Average Error: 27.2 → 2.6
Time: 19.7s
Precision: 64
Internal Precision: 576
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{{\left(\left(x \cdot cos\right) \cdot sin\right)}^{-2}}{\frac{1}{\cos \left(x \cdot 2\right)}}\]

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 27.2

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

  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 associate-/l*2.9

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

  7. Applied div-inv2.9

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

  8. Applied associate-/r*2.9

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

  10. Applied pow22.9

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

  11. Applied pow-flip2.6

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

  12. Simplified2.6

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

  13. Final simplification2.6

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

Runtime

Time bar (total: 19.7s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes2.62.61.80.80%
herbie shell --seed 2018339 +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))))