Average Error: 27.4 → 1.2
Time: 51.4s
Precision: 64
Internal Precision: 320
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{\left|\left(x \cdot cos\right) \cdot sin\right| \cdot \left|x \cdot \left(cos \cdot sin\right)\right|} \le -4.263919909618442 \cdot 10^{-232}:\\ \;\;\;\;\cos \left(2 \cdot x\right) \cdot \frac{1}{\left|\left(x \cdot cos\right) \cdot sin\right| \cdot \left|\left(x \cdot cos\right) \cdot sin\right|}\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{\left|\left(x \cdot cos\right) \cdot sin\right| \cdot \left|x \cdot \left(cos \cdot sin\right)\right|} \le 2.1752858719985863 \cdot 10^{-307}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{\left|cos \cdot \left(sin \cdot x\right)\right| \cdot \sqrt[3]{\left|cos \cdot \left(sin \cdot x\right)\right|}}}{{\left(\sqrt{\sqrt[3]{\left|cos \cdot \left(x \cdot sin\right)\right|}} \cdot \sqrt{\sqrt[3]{\left|cos \cdot \left(x \cdot sin\right)\right|}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left|\left|\left(sin \cdot cos\right) \cdot x\right|\right| \cdot \left|\left|\left(sin \cdot cos\right) \cdot x\right|\right|}\\ \end{array}\]

Error

Bits error versus x

Bits error versus cos

Bits error versus sin

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)) (* (fabs (* (* x cos) sin)) (fabs (* x (* cos sin))))) < -4.263919909618442e-232

    1. Initial program 44.8

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

    3. Applied add-sqr-sqrt44.9

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

    4. Applied simplify44.8

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

    5. Applied simplify0.7

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

    7. Applied div-inv0.7

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

    if -4.263919909618442e-232 < (/ (cos (* 2 x)) (* (fabs (* (* x cos) sin)) (fabs (* x (* cos sin))))) < 2.1752858719985863e-307

    1. Initial program 12.0

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

    3. Applied add-sqr-sqrt12.0

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

    4. Applied simplify12.0

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

    5. Applied simplify2.9

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

    6. Taylor expanded around 0 1.2

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

    8. Applied add-cube-cbrt1.4

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

    9. Applied unpow-prod-down1.4

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

    10. Applied associate-/r*0.8

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

    11. Applied simplify0.8

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

    13. Applied add-sqr-sqrt0.8

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

    if 2.1752858719985863e-307 < (/ (cos (* 2 x)) (* (fabs (* (* x cos) sin)) (fabs (* x (* cos sin)))))

    1. Initial program 43.9

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

    3. Applied add-sqr-sqrt44.0

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

    4. Applied simplify43.9

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

    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 simplify4.6

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

    9. Applied simplify2.0

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

Runtime

Time bar (total: 51.4s)Debug logProfile

herbie shell --seed 2018201 
(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))))