Average Error: 27.7 → 2.0
Time: 2.6m
Precision: 64
Internal Precision: 128
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 4.3286984011912626 \cdot 10^{-187}:\\ \;\;\;\;\left(\frac{\frac{1}{x}}{cos} \cdot \frac{1}{sin}\right) \cdot \frac{\frac{\cos \left(x \cdot 2\right)}{sin}}{cos \cdot x}\\ \mathbf{elif}\;x \le 1.5590888981364931 \cdot 10^{+156}:\\ \;\;\;\;\frac{1}{\frac{\left(x \cdot \left(cos \cdot sin\right)\right) \cdot \left(x \cdot \left(cos \cdot sin\right)\right)}{\cos \left(x \cdot 2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1}{x}}{cos} \cdot \frac{1}{sin}\right) \cdot \frac{\frac{\cos \left(x \cdot 2\right)}{sin}}{cos \cdot x}\\ \end{array}\]

Error

Bits error versus x

Bits error versus cos

Bits error versus sin

Derivation

  1. Split input into 2 regimes

  2. if x < 4.3286984011912626e-187 or 1.5590888981364931e+156 < x

    1. Initial program 28.2

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

    2. Simplified2.5

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

    4. Applied *-un-lft-identity2.5

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

    5. Applied times-frac2.3

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

    7. Applied associate-/r*2.3

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

    9. Applied add-cube-cbrt2.3

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{sin \cdot \left(x \cdot cos\right)} \cdot \frac{\frac{\cos \left(2 \cdot x\right)}{sin}}{x \cdot cos}\]

    10. Applied times-frac2.4

      \[\leadsto \color{blue}{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{sin} \cdot \frac{\sqrt[3]{1}}{x \cdot cos}\right)} \cdot \frac{\frac{\cos \left(2 \cdot x\right)}{sin}}{x \cdot cos}\]

    11. Simplified2.4

      \[\leadsto \left(\color{blue}{\frac{1}{sin}} \cdot \frac{\sqrt[3]{1}}{x \cdot cos}\right) \cdot \frac{\frac{\cos \left(2 \cdot x\right)}{sin}}{x \cdot cos}\]

    12. Simplified2.4

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

    if 4.3286984011912626e-187 < x < 1.5590888981364931e+156

    1. Initial program 26.3

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

    2. Simplified2.8

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

    4. Applied clear-num2.8

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

    5. Taylor expanded around inf 27.6

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

    6. Simplified0.9

      \[\leadsto \frac{1}{\frac{\color{blue}{\left(\left(cos \cdot sin\right) \cdot x\right) \cdot \left(\left(cos \cdot sin\right) \cdot x\right)}}{\cos \left(2 \cdot x\right)}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 4.3286984011912626 \cdot 10^{-187}:\\ \;\;\;\;\left(\frac{\frac{1}{x}}{cos} \cdot \frac{1}{sin}\right) \cdot \frac{\frac{\cos \left(x \cdot 2\right)}{sin}}{cos \cdot x}\\ \mathbf{elif}\;x \le 1.5590888981364931 \cdot 10^{+156}:\\ \;\;\;\;\frac{1}{\frac{\left(x \cdot \left(cos \cdot sin\right)\right) \cdot \left(x \cdot \left(cos \cdot sin\right)\right)}{\cos \left(x \cdot 2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1}{x}}{cos} \cdot \frac{1}{sin}\right) \cdot \frac{\frac{\cos \left(x \cdot 2\right)}{sin}}{cos \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019091 
(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))))