Average Error: 26.7 → 1.3
Time: 56.1s
Precision: 64
Internal Precision: 576
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{\left(\left|cos \cdot \left(x \cdot sin\right)\right|\right)}^{2}} \le 9.935743733159435 \cdot 10^{-295}:\\ \;\;\;\;\left(\left(\sqrt[3]{\frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{\left|\left(x \cdot cos\right) \cdot sin\right|}} \cdot \sqrt[3]{\frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{\left|\left(x \cdot cos\right) \cdot sin\right|}}\right) \cdot \sqrt[3]{\frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{\left|\left(x \cdot cos\right) \cdot sin\right|}}\right) \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\left|\left(x \cdot cos\right) \cdot sin\right|}\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{\left(\left|cos \cdot \left(x \cdot sin\right)\right|\right)}^{2}} \le 7.089544540157552 \cdot 10^{+297}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{{\left(\left|cos \cdot \left(x \cdot sin\right)\right|\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{\left|\left(x \cdot cos\right) \cdot sin\right|} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\left|x \cdot \left(cos \cdot sin\right)\right|}\\ \end{array}\]

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 3 regimes

  2. if (/ (cos (* 2 x)) (pow (fabs (* cos (* x sin))) 2)) < 9.935743733159435e-295

    1. Initial program 16.4

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

    3. Applied add-sqr-sqrt16.4

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

    4. Applied simplify16.4

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

    5. Applied simplify1.5

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

    7. Applied add-cube-cbrt1.7

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

    8. Applied times-frac1.2

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

    10. Applied add-cube-cbrt1.3

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

    if 9.935743733159435e-295 < (/ (cos (* 2 x)) (pow (fabs (* cos (* x sin))) 2)) < 7.089544540157552e+297

    1. Initial program 43.2

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

    3. Applied add-sqr-sqrt43.3

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

    4. Applied simplify43.2

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

    5. Applied simplify4.7

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

    6. Taylor expanded around 0 0.9

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

    if 7.089544540157552e+297 < (/ (cos (* 2 x)) (pow (fabs (* cos (* x sin))) 2))

    1. Initial program 52.3

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

    3. Applied add-sqr-sqrt52.3

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

    4. Applied simplify52.2

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

    5. Applied simplify5.1

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

    7. Applied add-cube-cbrt5.1

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

    8. Applied times-frac5.2

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

    10. Applied associate-*l*6.2

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

Runtime

Time bar (total: 56.1s)Debug logProfile

herbie shell --seed 2018214 +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))))