\[\cos \left(x + \varepsilon\right) - \cos x\]
Test:
NMSE problem 3.3.5
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 31.1 s
Input Error: 37.1
Output Error: 4.6
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left(\sin x \cdot \sin \varepsilon\right)}^2}{\cos x \cdot \cos \varepsilon + \sin x \cdot \sin \varepsilon} - \cos x & \text{when } \varepsilon \le -2.504876215430376 \cdot 10^{-23} \\ \left(\varepsilon \cdot \frac{1}{6}\right) \cdot {x}^3 - \varepsilon \cdot \left(\varepsilon \cdot \frac{1}{2} + x\right) & \text{when } \varepsilon \le 1.6258231275971248 \cdot 10^{-14} \\ \log \left(e^{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x}\right) & \text{otherwise} \end{cases}\)

    if eps < -2.504876215430376e-23

    1. Started with
      \[\cos \left(x + \varepsilon\right) - \cos x\]
      31.6
    2. Using strategy rm
      31.6
    3. Applied cos-sum to get
      \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
      3.7
    4. Using strategy rm
      3.7
    5. Applied flip-- to get
      \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x \leadsto \color{blue}{\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left(\sin x \cdot \sin \varepsilon\right)}^2}{\cos x \cdot \cos \varepsilon + \sin x \cdot \sin \varepsilon}} - \cos x\]
      3.8

    if -2.504876215430376e-23 < eps < 1.6258231275971248e-14

    1. Started with
      \[\cos \left(x + \varepsilon\right) - \cos x\]
      45.7
    2. Applied taylor to get
      \[\cos \left(x + \varepsilon\right) - \cos x \leadsto \frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)\]
      6.8
    3. Taylor expanded around 0 to get
      \[\color{red}{\frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)} \leadsto \color{blue}{\frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)}\]
      6.8
    4. Applied simplify to get
      \[\color{red}{\frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)} \leadsto \color{blue}{\left(\varepsilon \cdot \frac{1}{6}\right) \cdot {x}^3 - \varepsilon \cdot \left(\varepsilon \cdot \frac{1}{2} + x\right)}\]
      6.8

    if 1.6258231275971248e-14 < eps

    1. Started with
      \[\cos \left(x + \varepsilon\right) - \cos x\]
      30.8
    2. Using strategy rm
      30.8
    3. Applied cos-sum to get
      \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
      2.1
    4. Using strategy rm
      2.1
    5. Applied add-log-exp to get
      \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x} \leadsto \color{blue}{\log \left(e^{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x}\right)}\]
      2.3
  1. Removed slow pow expressions

Original test:


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