\[\cos \left(x + \varepsilon\right) - \cos x\]
Test:
NMSE problem 3.3.5
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 46.0 s
Input Error: 36.5
Output Error: 4.6
Log: ⚲
Profile: 🕒
\(\begin{cases} \log \left(e^{\cos x \cdot \cos \varepsilon}\right) - (\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_* & \text{when } \varepsilon \le -1.2532876517739886 \cdot 10^{-27} \\ \left(\varepsilon \cdot \frac{1}{6}\right) \cdot {x}^3 - \varepsilon \cdot (\frac{1}{2} * \varepsilon + x)_* & \text{when } \varepsilon \le 339.99651990139387 \\ \log \left(e^{\cos x \cdot \cos \varepsilon}\right) - (\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_* & \text{otherwise} \end{cases}\)

    if eps < -1.2532876517739886e-27 or 339.99651990139387 < eps

    1. Started with
      \[\cos \left(x + \varepsilon\right) - \cos x\]
      30.7
    2. Using strategy rm
      30.7
    3. Applied cos-sum to get
      \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
      2.5
    4. Applied associate--l- to get
      \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x} \leadsto \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]
      2.5
    5. Applied simplify to get
      \[\cos x \cdot \cos \varepsilon - \color{red}{\left(\sin x \cdot \sin \varepsilon + \cos x\right)} \leadsto \cos x \cdot \cos \varepsilon - \color{blue}{(\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*}\]
      2.5
    6. Using strategy rm
      2.5
    7. Applied add-log-exp to get
      \[\color{red}{\cos x \cdot \cos \varepsilon} - (\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_* \leadsto \color{blue}{\log \left(e^{\cos x \cdot \cos \varepsilon}\right)} - (\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*\]
      2.7

    if -1.2532876517739886e-27 < eps < 339.99651990139387

    1. Started with
      \[\cos \left(x + \varepsilon\right) - \cos x\]
      44.8
    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
      \[\color{red}{\frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)} \leadsto \color{blue}{\left(\varepsilon \cdot \frac{1}{6}\right) \cdot {x}^3 - \varepsilon \cdot (\frac{1}{2} * \varepsilon + x)_*}\]
      7.2
  1. Removed slow pow expressions

Original test:


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