Average Error: 27.0 → 1.3
Time: 1.2m
Precision: 64
Internal Precision: 576
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{\left|\left(cos \cdot sin\right) \cdot x\right|} \cdot \frac{\cos \left(x \cdot 2\right)}{\left|cos \cdot \left(sin \cdot x\right)\right|} \le 1.6261123667941 \cdot 10^{-313}:\\ \;\;\;\;\frac{\frac{\cos \left(2 \cdot x\right)}{\left|\left(x \cdot cos\right) \cdot sin\right|}}{\left|\left(x \cdot cos\right) \cdot sin\right|}\\ \mathbf{if}\;\frac{1}{\left|\left(cos \cdot sin\right) \cdot x\right|} \cdot \frac{\cos \left(x \cdot 2\right)}{\left|cos \cdot \left(sin \cdot x\right)\right|} \le 2.3999061954305667 \cdot 10^{+296}:\\ \;\;\;\;\frac{1}{\left|\left(cos \cdot sin\right) \cdot x\right|} \cdot \frac{\cos \left(x \cdot 2\right)}{\left|cos \cdot \left(sin \cdot x\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{\cos \left(2 \cdot x\right)}{\left|\left(x \cdot cos\right) \cdot sin\right|}} \cdot \sqrt{\frac{\cos \left(2 \cdot x\right)}{\left|\left(x \cdot cos\right) \cdot sin\right|}}}{\left|\left(x \cdot cos\right) \cdot sin\right|}\\ \end{array}\]

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 3 regimes

  2. if (* (/ 1 (fabs (* (* cos sin) x))) (/ (cos (* x 2)) (fabs (* cos (* sin x))))) < 1.6261123667941e-313

    1. Initial program 17.2

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

    3. Applied add-sqr-sqrt17.2

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

    4. Applied simplify17.2

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

    5. Applied simplify1.6

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

    7. Applied associate-/r*1.2

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

    if 1.6261123667941e-313 < (* (/ 1 (fabs (* (* cos sin) x))) (/ (cos (* x 2)) (fabs (* cos (* sin x))))) < 2.3999061954305667e+296

    1. Initial program 42.6

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

    3. Applied add-sqr-sqrt42.7

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

    4. Applied simplify42.6

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

    5. Applied simplify5.5

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

    7. Applied associate-/r*5.3

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

    9. Applied add-sqr-sqrt5.5

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

    10. Applied add-sqr-sqrt5.7

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

    11. Applied *-un-lft-identity5.7

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

    12. Applied times-frac5.7

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

    13. Applied times-frac5.7

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

    14. Applied simplify5.6

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

    15. Applied simplify1.1

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

    if 2.3999061954305667e+296 < (* (/ 1 (fabs (* (* cos sin) x))) (/ (cos (* x 2)) (fabs (* cos (* sin x)))))

    1. Initial program 55.9

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

    3. Applied add-sqr-sqrt55.9

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

    4. Applied simplify55.8

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

    5. Applied simplify5.0

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

    7. Applied associate-/r*5.1

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

    9. Applied add-sqr-sqrt5.2

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

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018193 +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))))