Average Error: 27.5 → 2.6
Time: 39.8s
Precision: 64
Internal Precision: 384
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{1}{cos \cdot \left|sin \cdot x\right|} \cdot \frac{\cos \left(2 \cdot x\right)}{\left(cos \cdot \sqrt{\left|sin \cdot x\right|}\right) \cdot \sqrt{\left|sin \cdot x\right|}}\]

Error

Bits error versus x

Bits error versus cos

Bits error versus sin

Derivation

  1. Initial program 27.5

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

  3. Applied add-sqr-sqrt27.5

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

  4. Applied simplify27.5

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

  5. Applied simplify19.4

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

  7. Applied pow219.4

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

  8. Applied pow-prod-down2.8

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

  10. Applied unpow22.8

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

  11. Applied *-un-lft-identity2.8

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

  12. Applied times-frac2.6

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

  14. Applied add-sqr-sqrt2.6

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

  15. Applied associate-*r*2.6

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

Runtime

Time bar (total: 39.8s)Debug logProfile

herbie shell --seed '#(1070258749 1877548225 2229079127 1588002776 3179087814 1886870650)' 
(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))))