\[\cos \left(x + \varepsilon\right) - \cos x\]
Test:
NMSE problem 3.3.5
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 25.6 s
Input Error: 36.6
Output Error: 3.6
Log: ⚲
Profile: 🕒
\(\begin{cases} (\left(\cos x\right) * \left(\cos \varepsilon\right) + \left(\left(-\sin x \cdot \sin \varepsilon\right) - \cos x\right))_* & \text{when } \varepsilon \le -8.102966659834604 \cdot 10^{-08} \\ \left(\varepsilon \cdot \frac{1}{6}\right) \cdot {x}^3 - \varepsilon \cdot (\frac{1}{2} * \varepsilon + x)_* & \text{when } \varepsilon \le 0.0009667373424041872 \\ (\left(\cos x\right) * \left(\cos \varepsilon\right) + \left(\left(-\sin x \cdot \sin \varepsilon\right) - \cos x\right))_* & \text{otherwise} \end{cases}\)

    if eps < -8.102966659834604e-08 or 0.0009667373424041872 < eps

    1. Started with
      \[\cos \left(x + \varepsilon\right) - \cos x\]
      29.7
    2. Using strategy rm
      29.7
    3. Applied cos-sum to get
      \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
      0.9
    4. Using strategy rm
      0.9
    5. Applied sub-neg to get
      \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon + \left(-\sin x \cdot \sin \varepsilon\right)\right)} - \cos x\]
      0.9
    6. Applied associate--l+ to get
      \[\color{red}{\left(\cos x \cdot \cos \varepsilon + \left(-\sin x \cdot \sin \varepsilon\right)\right) - \cos x} \leadsto \color{blue}{\cos x \cdot \cos \varepsilon + \left(\left(-\sin x \cdot \sin \varepsilon\right) - \cos x\right)}\]
      1.0
    7. Using strategy rm
      1.0
    8. Applied fma-def to get
      \[\color{red}{\cos x \cdot \cos \varepsilon + \left(\left(-\sin x \cdot \sin \varepsilon\right) - \cos x\right)} \leadsto \color{blue}{(\left(\cos x\right) * \left(\cos \varepsilon\right) + \left(\left(-\sin x \cdot \sin \varepsilon\right) - \cos x\right))_*}\]
      0.9

    if -8.102966659834604e-08 < eps < 0.0009667373424041872

    1. Started with
      \[\cos \left(x + \varepsilon\right) - \cos x\]
      46.4
    2. Applied taylor to get
      \[\cos \left(x + \varepsilon\right) - \cos x \leadsto \frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)\]
      7.3
    3. Taylor expanded around 0 to get
      \[\color{red}{\frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)} \leadsto \color{blue}{\frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)}\]
      7.3
    4. Applied simplify to get
      \[\color{red}{\frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)} \leadsto \color{blue}{\left(\varepsilon \cdot \frac{1}{6}\right) \cdot {x}^3 - \varepsilon \cdot (\frac{1}{2} * \varepsilon + x)_*}\]
      7.3
  1. Removed slow pow expressions

Original test:


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