Average Error: 27.2 → 2.5
Time: 41.7s
Precision: 64
Internal Precision: 576
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{\frac{\cos \left(x \cdot 2\right)}{\left|cos \cdot \left(sin \cdot x\right)\right|}}{{\left(\sqrt{\left|cos \cdot \left(x \cdot sin\right)\right|}\right)}^{2}}\]

Error

Bits error versus x

Bits error versus cos

Bits error versus sin

Derivation

  1. Initial program 27.2

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

  3. Applied add-sqr-sqrt27.2

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

  4. Applied simplify27.2

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

  5. Applied simplify2.9

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

  6. Taylor expanded around 0 2.7

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

  8. Applied add-sqr-sqrt2.9

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

  9. Applied unpow-prod-down2.9

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

  10. Applied associate-/r*2.6

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

  11. Applied simplify2.5

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

Runtime

Time bar (total: 41.7s)Debug logProfile

herbie shell --seed '#(1071725047 233389029 2036512464 3988615230 2972226563 1111574017)' 
(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))))