Average Error: 27.8 → 2.6
Time: 43.5s
Precision: 64
Internal Precision: 576
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{\left|\left|\left(sin \cdot cos\right) \cdot x\right|\right| \cdot \left|\left|\left(sin \cdot cos\right) \cdot x\right|\right|} \le 1.6580614592888172 \cdot 10^{+277}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left|\left|\left(sin \cdot cos\right) \cdot x\right|\right| \cdot \left|\left|\left(sin \cdot cos\right) \cdot x\right|\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\cos \left(x \cdot 2\right)} \cdot \frac{\sqrt{\cos \left(2 \cdot x\right)}}{{\left(\left|cos \cdot \left(x \cdot sin\right)\right|\right)}^{2}}\\ \end{array}\]

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)) (* (fabs (fabs (* (* sin cos) x))) (fabs (fabs (* (* sin cos) x))))) < 1.6580614592888172e+277

    1. Initial program 27.1

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

    3. Applied add-sqr-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 simplify27.1

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

    5. Applied simplify2.8

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

    7. Applied add-sqr-sqrt2.8

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

    8. Applied simplify4.4

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

    9. Applied simplify2.4

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

    if 1.6580614592888172e+277 < (/ (cos (* 2 x)) (* (fabs (fabs (* (* sin cos) x))) (fabs (fabs (* (* sin cos) x)))))

    1. Initial program 50.6

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

    3. Applied add-sqr-sqrt50.5

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

    4. Applied simplify50.2

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

    5. Applied simplify11.0

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

    6. Taylor expanded around 0 9.9

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

    8. Applied *-un-lft-identity9.9

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

    9. Applied unpow-prod-down9.9

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

    10. Applied add-sqr-sqrt9.9

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

    11. Applied times-frac9.9

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

    12. Applied simplify9.9

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

Runtime

Time bar (total: 43.5s)Debug logProfile

herbie shell --seed '#(1072107073 2127697367 3936270018 2300570620 2134894798 4023771849)' 
(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))))