Average Error: 27.2 → 7.1
Time: 50.8s
Precision: 64
Internal Precision: 384
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{cos \cdot \left(\left(\left(cos \cdot \left(x \cdot sin\right)\right) \cdot sin\right) \cdot x\right)} \le 0.0:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{cos \cdot \left(\left(\left(cos \cdot \left(x \cdot sin\right)\right) \cdot sin\right) \cdot x\right)}\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{cos \cdot \left(\left(\left(cos \cdot \left(x \cdot sin\right)\right) \cdot sin\right) \cdot x\right)} \le 1.7323185006285714 \cdot 10^{-297}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left(cos \cdot \left(cos \cdot \left(\left(x \cdot sin\right) \cdot sin\right)\right)\right) \cdot x}\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{cos \cdot \left(\left(\left(cos \cdot \left(x \cdot sin\right)\right) \cdot sin\right) \cdot x\right)} \le 5.537065305009333 \cdot 10^{+294}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{cos \cdot \left(\left(\left(cos \cdot \left(x \cdot sin\right)\right) \cdot sin\right) \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;(\frac{2}{3} \cdot \left(\frac{x}{sin \cdot cos} \cdot \frac{x}{sin \cdot cos}\right) + \left(\frac{\frac{1}{sin \cdot sin}}{\left(cos \cdot x\right) \cdot \left(cos \cdot x\right)}\right))_* - \frac{\frac{2}{sin \cdot cos}}{sin \cdot cos}\\ \end{array}\]

Error

Bits error versus x

Bits error versus cos

Bits error versus sin

Derivation

  1. Split input into 3 regimes

  2. if (/ (cos (* 2 x)) (* cos (* (* (* cos (* x sin)) sin) x))) < 0.0 or 1.7323185006285714e-297 < (/ (cos (* 2 x)) (* cos (* (* (* cos (* x sin)) sin) x))) < 5.537065305009333e+294

    1. Initial program 25.5

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

    3. Applied unpow225.5

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

    4. Applied associate-*l*21.1

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

    6. Applied unpow221.1

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

    7. Applied associate-*r*14.1

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

    9. Applied associate-*r*11.0

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

    11. Applied associate-*r*5.4

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

    if 0.0 < (/ (cos (* 2 x)) (* cos (* (* (* cos (* x sin)) sin) x))) < 1.7323185006285714e-297

    1. Initial program 28.6

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

    3. Applied unpow228.6

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

    4. Applied associate-*l*7.6

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

    6. Applied unpow27.6

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

    7. Applied associate-*r*3.5

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

    9. Applied associate-*r*2.5

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

    11. Applied associate-*r*36.9

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

    if 5.537065305009333e+294 < (/ (cos (* 2 x)) (* cos (* (* (* cos (* x sin)) sin) x)))

    1. Initial program 58.0

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

    2. Taylor expanded around 0 56.7

      \[\leadsto \color{blue}{\left(\frac{2}{3} \cdot \frac{{x}^{2}}{{sin}^{2} \cdot {cos}^{2}} + \frac{1}{{sin}^{2} \cdot \left({x}^{2} \cdot {cos}^{2}\right)}\right) - 2 \cdot \frac{1}{{cos}^{2} \cdot {sin}^{2}}}\]

    3. Applied simplify36.1

      \[\leadsto \color{blue}{(\frac{2}{3} \cdot \left(\frac{x}{sin \cdot cos} \cdot \frac{x}{sin \cdot cos}\right) + \left(\frac{\frac{1}{sin \cdot sin}}{\left(cos \cdot x\right) \cdot \left(cos \cdot x\right)}\right))_* - \frac{\frac{2}{sin \cdot cos}}{sin \cdot cos}}\]
  3. Recombined 3 regimes into one program.

Runtime

Time bar (total: 50.8s)Debug logProfile

herbie shell --seed '#(1070100504 930361288 1279167582 284574201 1450237281 2578255382)' +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))))