Average Error: 28.1 → 1.8
Time: 45.6s
Precision: 64
Internal Precision: 384
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{\left|\left|\left(sin \cdot cos\right) \cdot x\right|\right| \cdot \left|\left|\left(sin \cdot cos\right) \cdot x\right|\right|} \le 7.259820169537017 \cdot 10^{-55}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\left|\left(x \cdot cos\right) \cdot sin\right|}} \cdot \sqrt{\frac{1}{\left|\left(x \cdot cos\right) \cdot sin\right|}}\right) \cdot \frac{\cos \left(2 \cdot x\right)}{\left|\left(x \cdot cos\right) \cdot sin\right|}\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{\left|\left|\left(sin \cdot cos\right) \cdot x\right|\right| \cdot \left|\left|\left(sin \cdot cos\right) \cdot x\right|\right|} \le 1.0565231476000496 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{\left|\left(cos \cdot sin\right) \cdot x\right|}}{\left|\left(cos \cdot sin\right) \cdot x\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{\left|\left(x \cdot cos\right) \cdot sin\right|} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\left|\left(x \cdot cos\right) \cdot sin\right|}\\ \end{array}\]

Error

Bits error versus x

Bits error versus cos

Bits error versus sin

Derivation

  1. Split input into 3 regimes

  2. if (/ (cos (* 2 x)) (* (fabs (fabs (* (* sin cos) x))) (fabs (fabs (* (* sin cos) x))))) < 7.259820169537017e-55

    1. Initial program 22.9

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

    3. Applied add-sqr-sqrt22.9

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

    4. Applied simplify22.9

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

    5. Applied simplify2.3

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

    7. Applied *-un-lft-identity2.3

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

    8. Applied times-frac2.0

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

    10. Applied add-sqr-sqrt2.0

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

    if 7.259820169537017e-55 < (/ (cos (* 2 x)) (* (fabs (fabs (* (* sin cos) x))) (fabs (fabs (* (* sin cos) x))))) < 1.0565231476000496e+305

    1. Initial program 43.6

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

    3. Applied add-sqr-sqrt43.6

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

    4. Applied simplify43.5

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

    5. Applied simplify5.0

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

    7. Applied *-un-lft-identity5.0

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

    8. Applied times-frac5.0

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

    9. Taylor expanded around inf 5.0

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

    10. Applied simplify0.9

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

    if 1.0565231476000496e+305 < (/ (cos (* 2 x)) (* (fabs (fabs (* (* sin cos) x))) (fabs (fabs (* (* sin cos) x)))))

    1. Initial program 57.5

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

    3. Applied add-sqr-sqrt57.6

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

    4. Applied simplify57.6

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

    5. Applied simplify2.1

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

    7. Applied add-cube-cbrt2.7

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

    8. Applied times-frac2.7

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

Runtime

Time bar (total: 45.6s)Debug logProfile

herbie shell --seed '#(1070706311 3771791028 4128836681 4194990999 2341756049 504035650)' +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))))