Average Error: 27.9 → 1.2
Time: 39.4s
Precision: 64
Internal Precision: 576
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\cos x \cdot \cos x - \sin x \cdot \sin x}{\left|cos \cdot \left(x \cdot sin\right)\right| \cdot \left|cos \cdot \left(x \cdot sin\right)\right|} \le 6.5631148755239296 \cdot 10^{-298}:\\ \;\;\;\;\frac{1}{\left|\left(cos \cdot x\right) \cdot sin\right|} \cdot \frac{\cos \left(2 \cdot x\right)}{\left|sin \cdot \left(x \cdot cos\right)\right|}\\ \mathbf{if}\;\frac{\cos x \cdot \cos x - \sin x \cdot \sin x}{\left|cos \cdot \left(x \cdot sin\right)\right| \cdot \left|cos \cdot \left(x \cdot sin\right)\right|} \le 5.115423973217656 \cdot 10^{+302}:\\ \;\;\;\;\frac{\cos x \cdot \cos x - \sin x \cdot \sin x}{\left|cos \cdot \left(x \cdot sin\right)\right| \cdot \left|cos \cdot \left(x \cdot sin\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(2 \cdot x\right)}{\left|sin \cdot \left(x \cdot cos\right)\right|}}{{\left(\sqrt{\left|x \cdot \left(cos \cdot sin\right)\right|}\right)}^{2}}\\ \end{array}\]

Error

Bits error versus x

Bits error versus cos

Bits error versus sin

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if (/ (- (* (cos x) (cos x)) (* (sin x) (sin x))) (* (fabs (* cos (* x sin))) (fabs (* cos (* x sin))))) < 6.5631148755239296e-298

    1. Initial program 17.2

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

    3. Applied add-sqr-sqrt17.3

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

    4. Applied simplify17.2

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

    5. Applied simplify2.8

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

    6. Taylor expanded around 0 1.7

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

    8. Applied add-sqr-sqrt1.8

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

    9. Applied unpow-prod-down1.8

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

    10. Applied *-un-lft-identity1.8

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

    11. Applied times-frac1.4

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

    12. Applied simplify2.4

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

    13. Applied simplify1.3

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

    if 6.5631148755239296e-298 < (/ (- (* (cos x) (cos x)) (* (sin x) (sin x))) (* (fabs (* cos (* x sin))) (fabs (* cos (* x sin))))) < 5.115423973217656e+302

    1. Initial program 44.0

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

    3. Applied add-sqr-sqrt44.1

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

    4. Applied simplify44.0

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

    5. Applied simplify0.9

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

    7. Applied cos-21.0

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

    if 5.115423973217656e+302 < (/ (- (* (cos x) (cos x)) (* (sin x) (sin x))) (* (fabs (* cos (* x sin))) (fabs (* cos (* x sin)))))

    1. Initial program 58.3

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

    3. Applied add-sqr-sqrt58.4

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

    4. Applied simplify58.4

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

    5. Applied simplify48.6

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

    6. Taylor expanded around 0 1.5

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

    8. Applied add-sqr-sqrt1.9

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

    9. Applied unpow-prod-down1.9

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

    10. Applied associate-/r*1.9

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

    11. Applied simplify3.4

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

Runtime

Time bar (total: 39.4s)Debug logProfile

herbie shell --seed 2019053 +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))))