Average Error: 27.3 → 1.6
Time: 48.6s
Precision: 64
Internal Precision: 384
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{\cos \left(2 \cdot x\right)}}{\frac{\left|\left|\left(sin \cdot cos\right) \cdot x\right|\right| \cdot \left|\left|\left(sin \cdot cos\right) \cdot x\right|\right|}{\sqrt{\cos \left(2 \cdot x\right)}}} \le 1.3269981157202407 \cdot 10^{-266}:\\ \;\;\;\;\frac{1}{\left|\left(x \cdot cos\right) \cdot sin\right|} \cdot \left(\sqrt{\frac{\cos \left(2 \cdot x\right)}{\left|\left(x \cdot cos\right) \cdot sin\right|}} \cdot \sqrt{\frac{\cos \left(2 \cdot x\right)}{\left|\left(x \cdot cos\right) \cdot sin\right|}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left|\left|\left(sin \cdot cos\right) \cdot x\right|\right| \cdot \left|\left|\left(sin \cdot cos\right) \cdot x\right|\right|}\\ \end{array}\]

Error

Bits error versus x

Bits error versus cos

Bits error versus sin

Derivation

  1. Split input into 2 regimes

  2. if (/ (sqrt (cos (* 2 x))) (/ (* (fabs (fabs (* (* sin cos) x))) (fabs (fabs (* (* sin cos) x)))) (sqrt (cos (* 2 x))))) < 1.3269981157202407e-266

    1. Initial program 11.8

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

    3. Applied add-sqr-sqrt11.8

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

    4. Applied simplify11.8

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

    5. Applied simplify1.2

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

    7. Applied *-un-lft-identity1.2

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

    8. Applied times-frac0.5

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

    10. Applied add-sqr-sqrt0.5

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

    if 1.3269981157202407e-266 < (/ (sqrt (cos (* 2 x))) (/ (* (fabs (fabs (* (* sin cos) x))) (fabs (fabs (* (* sin cos) x)))) (sqrt (cos (* 2 x)))))

    1. Initial program 34.6

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

    3. Applied add-sqr-sqrt34.6

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

    4. Applied simplify34.6

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

    5. Applied simplify3.6

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

    7. Applied add-sqr-sqrt3.6

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

    8. Applied simplify4.9

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

    9. Applied simplify2.1

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

Runtime

Time bar (total: 48.6s)Debug logProfile

herbie shell --seed '#(1070833653 108281690 3330367898 3632331308 3494323072 43156186)' +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))))