Average Error: 27.2 → 1.8
Time: 1.1m
Precision: 64
Internal Precision: 576
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\sqrt{\frac{\cos \left(2 \cdot x\right)}{{\left(\left|cos \cdot \left(x \cdot sin\right)\right|\right)}^{2}}} \cdot \sqrt{\frac{\cos \left(2 \cdot x\right)}{{\left(\left|cos \cdot \left(x \cdot sin\right)\right|\right)}^{2}}} \le 8.426925871894354 \cdot 10^{-297}:\\ \;\;\;\;\frac{\sqrt[3]{\cos \left(x \cdot 2\right)}}{\frac{\left|x \cdot \left(sin \cdot cos\right)\right|}{\sqrt[3]{\cos \left(x \cdot 2\right)}}} \cdot \frac{\sqrt[3]{\cos \left(x \cdot 2\right)}}{\left|\left(cos \cdot x\right) \cdot sin\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{\left|cos \cdot \left(x \cdot sin\right)\right|} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\left|cos \cdot \left(x \cdot sin\right)\right|}\\ \end{array}\]

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 (* (sqrt (/ (cos (* 2 x)) (pow (fabs (* cos (* x sin))) 2))) (sqrt (/ (cos (* 2 x)) (pow (fabs (* cos (* x sin))) 2)))) < 8.426925871894354e-297

    1. Initial program 9.8

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

    3. Applied add-sqr-sqrt9.8

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

    4. Applied simplify9.8

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

    5. Applied simplify0.9

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

    6. Taylor expanded around 0 3.2

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

    8. Applied add-sqr-sqrt3.2

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

    9. Applied unpow-prod-down3.2

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

    10. Applied add-cube-cbrt3.2

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

    11. Applied times-frac2.6

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

    12. Applied simplify2.8

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

    13. Applied simplify0.5

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

    if 8.426925871894354e-297 < (* (sqrt (/ (cos (* 2 x)) (pow (fabs (* cos (* x sin))) 2))) (sqrt (/ (cos (* 2 x)) (pow (fabs (* cos (* x sin))) 2))))

    1. Initial program 44.2

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

    3. Applied add-sqr-sqrt44.2

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

    4. Applied simplify44.1

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

    5. Applied simplify4.6

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

    6. 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 add-cube-cbrt3.0

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

    10. Applied times-frac3.0

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

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed 2018199 
(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))))