\[\cos \left(x + \varepsilon\right) - \cos x\]
Test:
NMSE problem 3.3.5
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 26.7 s
Input Error: 37.3
Output Error: 5.6
Log: ⚲
Profile: 🕒
\(\begin{cases} \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x & \text{when } \varepsilon \le -8.852571138183384 \cdot 10^{-52} \\ \left({x}^3 \cdot \frac{1}{6} - (\frac{1}{2} * \varepsilon + x)_*\right) \cdot \varepsilon & \text{when } \varepsilon \le 17.12856537901564 \\ \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x & \text{otherwise} \end{cases}\)

    if eps < -8.852571138183384e-52 or 17.12856537901564 < eps

    1. Started with
      \[\cos \left(x + \varepsilon\right) - \cos x\]
      32.8
    2. Using strategy rm
      32.8
    3. Applied cos-sum to get
      \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
      4.6

    if -8.852571138183384e-52 < eps < 17.12856537901564

    1. Started with
      \[\cos \left(x + \varepsilon\right) - \cos x\]
      44.5
    2. Applied taylor to get
      \[\cos \left(x + \varepsilon\right) - \cos x \leadsto \frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)\]
      7.1
    3. Taylor expanded around 0 to get
      \[\color{red}{\frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)} \leadsto \color{blue}{\frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)}\]
      7.1
    4. Applied simplify to get
      \[\color{red}{\frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)} \leadsto \color{blue}{\left(\varepsilon \cdot \frac{1}{6}\right) \cdot {x}^3 - \varepsilon \cdot (\frac{1}{2} * \varepsilon + x)_*}\]
      7.1
    5. Applied taylor to get
      \[\left(\varepsilon \cdot \frac{1}{6}\right) \cdot {x}^3 - \varepsilon \cdot (\frac{1}{2} * \varepsilon + x)_* \leadsto \varepsilon \cdot \left(\frac{1}{6} \cdot {x}^3 - (\frac{1}{2} * \varepsilon + x)_*\right)\]
      7.1
    6. Taylor expanded around 0 to get
      \[\color{red}{\varepsilon \cdot \left(\frac{1}{6} \cdot {x}^3 - (\frac{1}{2} * \varepsilon + x)_*\right)} \leadsto \color{blue}{\varepsilon \cdot \left(\frac{1}{6} \cdot {x}^3 - (\frac{1}{2} * \varepsilon + x)_*\right)}\]
      7.1
    7. Applied simplify to get
      \[\varepsilon \cdot \left(\frac{1}{6} \cdot {x}^3 - (\frac{1}{2} * \varepsilon + x)_*\right) \leadsto \left({x}^3 \cdot \frac{1}{6} - (\frac{1}{2} * \varepsilon + x)_*\right) \cdot \varepsilon\]
      7.1
    8. Applied final simplification
  1. Removed slow pow expressions

Original test:


(lambda ((x default) (eps default))
  #:name "NMSE problem 3.3.5"
  (- (cos (+ x eps)) (cos x)))