Average Error: 27.0 → 2.3
Time: 52.5s
Precision: 64
Internal Precision: 320
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;sin \le 1.1414537332208317 \cdot 10^{-291} \lor \neg \left(sin \le 5.922583641732854 \cdot 10^{+171}\right):\\ \;\;\;\;\frac{\frac{\cos \left(2 \cdot x\right)}{\left|\left(sin \cdot x\right) \cdot cos\right|}}{\left|\left(sin \cdot x\right) \cdot cos\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left|\left(cos \cdot x\right) \cdot sin\right|} \cdot \frac{\cos \left(2 \cdot x\right)}{\left|\left(cos \cdot x\right) \cdot sin\right|}\\ \end{array}\]

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 2 regimes

  2. if sin < 1.1414537332208317e-291 or 5.922583641732854e+171 < sin

    1. Initial program 26.8

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

    3. Applied add-sqr-sqrt26.8

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

    4. Applied simplify26.8

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

    5. Applied simplify3.2

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

    6. Taylor expanded around 0 2.7

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

    8. Applied unpow22.7

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

    9. Applied associate-/r*2.4

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

    if 1.1414537332208317e-291 < sin < 5.922583641732854e+171

    1. Initial program 27.4

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

    3. Applied add-sqr-sqrt27.5

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

    4. Applied simplify27.4

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

    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

      \[\leadsto \color{blue}{\frac{1}{\left|\left(x \cdot cos\right) \cdot sin\right|} \cdot \frac{\cos \left(2 \cdot x\right)}{\left|\left(x \cdot cos\right) \cdot sin\right|}}\]
  3. Recombined 2 regimes into one program.

  4. Applied simplify2.3

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;sin \le 1.1414537332208317 \cdot 10^{-291} \lor \neg \left(sin \le 5.922583641732854 \cdot 10^{+171}\right):\\ \;\;\;\;\frac{\frac{\cos \left(2 \cdot x\right)}{\left|\left(sin \cdot x\right) \cdot cos\right|}}{\left|\left(sin \cdot x\right) \cdot cos\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left|\left(cos \cdot x\right) \cdot sin\right|} \cdot \frac{\cos \left(2 \cdot x\right)}{\left|\left(cos \cdot x\right) \cdot sin\right|}\\ \end{array}}\]

Runtime

Time bar (total: 52.5s)Debug logProfile

herbie shell --seed 2018207 
(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))))