Average Error: 27.4 → 0.9
Time: 47.5s
Precision: 64
Internal Precision: 384
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)} \le +\infty:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\cos \left(x \cdot 2\right)} \cdot \sqrt[3]{\cos \left(x \cdot 2\right)}}{1} \cdot \frac{\sqrt[3]{\cos \left(x \cdot 2\right)}}{\left|cos \cdot \left(sin \cdot x\right)\right|}}{{\left(\sqrt{\left|cos \cdot \left(x \cdot sin\right)\right|}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left|\left|\left(sin \cdot cos\right) \cdot x\right|\right| \cdot \left|\left|\left(sin \cdot cos\right) \cdot x\right|\right|}\\ \end{array}\]

Error

Bits error versus x

Bits error versus cos

Bits error versus sin

Derivation

  1. Split input into 2 regimes

  2. if (/ (cos (* 2 x)) (* (pow cos 2) (* (* x (pow sin 2)) x))) < +inf.0

    1. Initial program 18.2

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

    3. Applied add-sqr-sqrt18.2

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

    4. Applied simplify18.1

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

    5. Applied simplify1.7

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

    6. Taylor expanded around 0 0.6

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

    8. Applied add-sqr-sqrt0.8

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

    9. Applied unpow-prod-down0.8

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

    10. Applied associate-/r*0.5

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

    11. Applied simplify0.4

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

    13. Applied *-un-lft-identity0.4

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

    14. Applied add-cube-cbrt0.5

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

    15. Applied times-frac0.5

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

    if +inf.0 < (/ (cos (* 2 x)) (* (pow cos 2) (* (* x (pow sin 2)) x)))

    1. Initial program 62.1

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

    3. Applied add-sqr-sqrt62.1

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

    4. Applied simplify62.1

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

    5. Applied simplify7.0

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

    7. Applied add-sqr-sqrt7.0

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

    8. Applied simplify8.3

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

    9. Applied simplify2.5

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

Runtime

Time bar (total: 47.5s)Debug logProfile

herbie shell --seed '#(1070609872 3456127585 2380521889 2328837196 1765472538 734540918)' 
(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))))