Average Error: 28.2 → 7.8
Time: 17.5s
Precision: 64
Internal Precision: 128
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;{cos}^{2} \le 3.1213865891969343 \cdot 10^{-187} \lor \neg \left({cos}^{2} \le 1.182816563802459 \cdot 10^{+83}\right):\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{\left(x \cdot \left(sin \cdot \left(\left(sin \cdot x\right) \cdot cos\right)\right)\right) \cdot cos}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{cos \cdot \left(cos \cdot \left(\left(sin \cdot x\right) \cdot \left(sin \cdot x\right)\right)\right)}\\ \end{array}\]

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. Split input into 2 regimes

  2. if (pow cos 2) < 3.1213865891969343e-187 or 1.182816563802459e+83 < (pow cos 2)

    1. Initial program 30.6

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

    3. Applied unpow230.6

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

    4. Applied associate-*l*24.9

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

    6. Applied unpow224.9

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

    7. Applied associate-*r*18.9

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

    9. Applied associate-*r*15.1

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

    11. Applied associate-*r*8.8

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

    if 3.1213865891969343e-187 < (pow cos 2) < 1.182816563802459e+83

    1. Initial program 19.6

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

    3. Applied unpow219.6

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

    4. Applied associate-*l*19.6

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

    6. Applied unpow219.6

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

    7. Applied associate-*r*9.0

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

    9. Applied associate-*l*4.5

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{cos \cdot \left(cos \cdot \color{blue}{\left(\left(x \cdot sin\right) \cdot \left(sin \cdot x\right)\right)}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification7.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;{cos}^{2} \le 3.1213865891969343 \cdot 10^{-187} \lor \neg \left({cos}^{2} \le 1.182816563802459 \cdot 10^{+83}\right):\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{\left(x \cdot \left(sin \cdot \left(\left(sin \cdot x\right) \cdot cos\right)\right)\right) \cdot cos}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{cos \cdot \left(cos \cdot \left(\left(sin \cdot x\right) \cdot \left(sin \cdot x\right)\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019021 
(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))))