Average Error: 27.6 → 2.6
Time: 35.3s
Precision: 64
Internal Precision: 128
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;cos \le 4.453024777768609 \cdot 10^{-148}:\\ \;\;\;\;\left(\sqrt[3]{\frac{\frac{\cos \left(2 \cdot x\right)}{sin \cdot \left(x \cdot cos\right)}}{sin \cdot \left(x \cdot cos\right)}} \cdot \sqrt[3]{\frac{\frac{\cos \left(2 \cdot x\right)}{sin \cdot \left(x \cdot cos\right)}}{sin \cdot \left(x \cdot cos\right)}}\right) \cdot \sqrt[3]{\frac{\frac{\cos \left(2 \cdot x\right)}{sin \cdot \left(x \cdot cos\right)}}{sin \cdot \left(x \cdot cos\right)}}\\ \mathbf{elif}\;cos \le 1.1653858133682418 \cdot 10^{+157}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left(cos \cdot \left(sin \cdot x\right)\right) \cdot \left(cos \cdot \left(sin \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{sin}}{x \cdot cos}}{sin \cdot \left(x \cdot cos\right)}} \cdot \sqrt{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{sin}}{x \cdot cos}}{sin \cdot \left(x \cdot cos\right)}}\\ \end{array}\]

Error

Bits error versus x

Bits error versus cos

Bits error versus sin

Derivation

  1. Split input into 3 regimes

  2. if cos < 4.453024777768609e-148

    1. Initial program 32.2

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

    2. Simplified3.0

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

    4. Applied associate-/r*2.7

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

    6. Applied add-cube-cbrt3.1

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

    if 4.453024777768609e-148 < cos < 1.1653858133682418e+157

    1. Initial program 19.7

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

    2. Simplified2.2

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

    3. Taylor expanded around -inf 24.1

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

    4. Simplified0.7

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

    if 1.1653858133682418e+157 < cos

    1. Initial program 23.9

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

    2. Simplified2.7

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

    4. Applied associate-/r*2.3

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

    6. Applied associate-/r*2.3

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

    8. Applied add-sqr-sqrt3.7

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

  4. Final simplification2.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;cos \le 4.453024777768609 \cdot 10^{-148}:\\ \;\;\;\;\left(\sqrt[3]{\frac{\frac{\cos \left(2 \cdot x\right)}{sin \cdot \left(x \cdot cos\right)}}{sin \cdot \left(x \cdot cos\right)}} \cdot \sqrt[3]{\frac{\frac{\cos \left(2 \cdot x\right)}{sin \cdot \left(x \cdot cos\right)}}{sin \cdot \left(x \cdot cos\right)}}\right) \cdot \sqrt[3]{\frac{\frac{\cos \left(2 \cdot x\right)}{sin \cdot \left(x \cdot cos\right)}}{sin \cdot \left(x \cdot cos\right)}}\\ \mathbf{elif}\;cos \le 1.1653858133682418 \cdot 10^{+157}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left(cos \cdot \left(sin \cdot x\right)\right) \cdot \left(cos \cdot \left(sin \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{sin}}{x \cdot cos}}{sin \cdot \left(x \cdot cos\right)}} \cdot \sqrt{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{sin}}{x \cdot cos}}{sin \cdot \left(x \cdot cos\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019051 
(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))))