Average Error: 27.3 → 1.3
Time: 1.1m
Precision: 64
Internal Precision: 576
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{\sqrt[3]{{\left(\left(\left(x \cdot cos\right) \cdot \left(sin \cdot sin\right)\right) \cdot \left(x \cdot cos\right)\right)}^{3}}} \le 4.894433632576957 \cdot 10^{-86}:\\ \;\;\;\;\frac{1}{\left|\left(cos \cdot x\right) \cdot sin\right|} \cdot \frac{\cos \left(2 \cdot x\right)}{\left|sin \cdot \left(x \cdot cos\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{{\left(\left|x \cdot \left(cos \cdot sin\right)\right|\right)}^{2}}\\ \end{array}\]

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 (/ (cos (* 2 x)) (cbrt (pow (* (* (* x cos) (* sin sin)) (* x cos)) 3))) < 4.894433632576957e-86

    1. Initial program 21.3

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

    3. Applied add-sqr-sqrt21.3

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

    4. Applied simplify21.3

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

    5. Applied simplify2.1

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

    6. Taylor expanded around inf 2.9

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

    8. Applied add-sqr-sqrt3.0

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

    9. Applied unpow-prod-down3.0

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

    10. Applied *-un-lft-identity3.0

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

    11. Applied times-frac2.7

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

    12. Applied simplify2.8

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

    13. Applied simplify0.5

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

    if 4.894433632576957e-86 < (/ (cos (* 2 x)) (cbrt (pow (* (* (* x cos) (* sin sin)) (* x cos)) 3)))

    1. Initial program 43.5

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

    3. Applied add-sqr-sqrt43.5

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

    4. Applied simplify43.5

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

    5. Applied simplify4.1

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

    6. Taylor expanded around inf 3.6

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

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed 2018296 +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))))