Average Error: 27.4 → 2.5
Time: 35.9s
Precision: 64
Internal Precision: 384
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{1}{\left|\sqrt{\left|cos\right|} \cdot \left(\sqrt{\left|cos\right|} \cdot \left(x \cdot sin\right)\right)\right|} \cdot \frac{\cos \left(2 \cdot x\right)}{\left|\left|cos\right| \cdot \left(x \cdot sin\right)\right|}\]

Error

Bits error versus x

Bits error versus cos

Bits error versus sin

Derivation

  1. Initial program 27.4

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

  3. Applied add-sqr-sqrt27.4

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

  4. Applied associate-*l*27.4

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

  5. Applied simplify19.3

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

  7. Applied add-sqr-sqrt19.3

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

  8. Applied simplify19.3

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

  9. Applied simplify2.6

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

  11. Applied *-un-lft-identity2.6

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

  12. Applied times-frac2.4

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

  14. Applied add-sqr-sqrt2.5

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

  15. Applied associate-*l*2.5

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

Runtime

Time bar (total: 35.9s)Debug logProfile

herbie shell --seed '#(1064269945 2896236262 301053905 1701069080 1701464310 1614783279)' 
(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))))