\[\cos \left(x + \varepsilon\right) - \cos x\]
Test:
NMSE problem 3.3.5
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 9.1 s
Input Error: 17.9
Output Error: 3.4
Log: ⚲
Profile: 🕒
\(\begin{cases} \left(\cos x \cdot \cos \varepsilon - (e^{\log_* (1 + \sin x \cdot \sin \varepsilon)} - 1)^*\right) - \cos x & \text{when } \varepsilon \le -2.6690726f-17 \\ \left(\varepsilon \cdot \frac{1}{6}\right) \cdot {x}^3 - \varepsilon \cdot (\frac{1}{2} * \varepsilon + x)_* & \text{when } \varepsilon \le 0.028850537f0 \\ \left(\cos x \cdot \cos \varepsilon - (e^{\log_* (1 + \sin x \cdot \sin \varepsilon)} - 1)^*\right) - \cos x & \text{otherwise} \end{cases}\)

    if eps < -2.6690726f-17 or 0.028850537f0 < eps

    1. Started with
      \[\cos \left(x + \varepsilon\right) - \cos x\]
      16.6
    2. Using strategy rm
      16.6
    3. Applied cos-sum to get
      \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
      3.2
    4. Using strategy rm
      3.2
    5. Applied expm1-log1p-u to get
      \[\left(\cos x \cdot \cos \varepsilon - \color{red}{\sin x \cdot \sin \varepsilon}\right) - \cos x \leadsto \left(\cos x \cdot \cos \varepsilon - \color{blue}{(e^{\log_* (1 + \sin x \cdot \sin \varepsilon)} - 1)^*}\right) - \cos x\]
      3.2

    if -2.6690726f-17 < eps < 0.028850537f0

    1. Started with
      \[\cos \left(x + \varepsilon\right) - \cos x\]
      20.7
    2. Applied taylor to get
      \[\cos \left(x + \varepsilon\right) - \cos x \leadsto \frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)\]
      3.8
    3. Taylor expanded around 0 to get
      \[\color{red}{\frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)} \leadsto \color{blue}{\frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)}\]
      3.8
    4. Applied simplify to get
      \[\color{red}{\frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)} \leadsto \color{blue}{\left(\varepsilon \cdot \frac{1}{6}\right) \cdot {x}^3 - \varepsilon \cdot (\frac{1}{2} * \varepsilon + x)_*}\]
      3.8
  1. Removed slow pow expressions

Original test:


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