Average Error: 27.3 → 2.5
Time: 42.2s
Precision: 64
Internal Precision: 384
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{\frac{\cos \left(x + x\right)}{\left|cos\right| \cdot \left(sin \cdot x\right)}}{\sqrt{\left|cos\right|} \cdot \left(\sqrt{\left|cos\right|} \cdot \left(sin \cdot x\right)\right)}\]

Error

Bits error versus x

Bits error versus cos

Bits error versus sin

Derivation

  1. Initial program 27.3

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

  3. Applied add-sqr-sqrt27.3

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

  4. Applied associate-*l*27.3

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

  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. Taylor expanded around -inf 62.5

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

  7. Applied simplify2.5

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

  9. Applied add-sqr-sqrt2.5

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

  10. Applied associate-*l*2.5

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

Runtime

Time bar (total: 42.2s)Debug logProfile

herbie shell --seed '#(1070355188 2193211668 3977393919 3454156579 3755371326 1656365382)' +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))))