\[\cos \left(x + \varepsilon\right) - \cos x\]
Test:
NMSE problem 3.3.5
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 14.3 s
Input Error: 18.1
Output Error: 3.7
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{\frac{{\left({\left(\cos x \cdot \cos \varepsilon\right)}^2\right)}^2 - {\left({\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2\right)}^2}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + {\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2}}{\cos x \cdot \cos \varepsilon + \left(\sin x \cdot \sin \varepsilon + \cos x\right)} & \text{when } \varepsilon \le -2.474651f-15 \\ \left(\varepsilon \cdot \frac{1}{6}\right) \cdot {x}^3 - \varepsilon \cdot \left(\varepsilon \cdot \frac{1}{2} + x\right) & \text{when } \varepsilon \le 2.1195103f-10 \\ \frac{\frac{{\left({\left(\cos x \cdot \cos \varepsilon\right)}^2\right)}^2 - {\left({\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2\right)}^2}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + {\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2}}{\cos x \cdot \cos \varepsilon + \left(\sin x \cdot \sin \varepsilon + \cos x\right)} & \text{otherwise} \end{cases}\)

    if eps < -2.474651f-15

    1. Started with
      \[\cos \left(x + \varepsilon\right) - \cos x\]
      17.9
    2. Using strategy rm
      17.9
    3. Applied cos-sum to get
      \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
      4.3
    4. Applied associate--l- to get
      \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x} \leadsto \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]
      4.3
    5. Using strategy rm
      4.3
    6. Applied flip-- to get
      \[\color{red}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)} \leadsto \color{blue}{\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2}{\cos x \cdot \cos \varepsilon + \left(\sin x \cdot \sin \varepsilon + \cos x\right)}}\]
      4.3
    7. Using strategy rm
      4.3
    8. Applied flip-- to get
      \[\frac{\color{red}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2}}{\cos x \cdot \cos \varepsilon + \left(\sin x \cdot \sin \varepsilon + \cos x\right)} \leadsto \frac{\color{blue}{\frac{{\left({\left(\cos x \cdot \cos \varepsilon\right)}^2\right)}^2 - {\left({\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2\right)}^2}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + {\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2}}}{\cos x \cdot \cos \varepsilon + \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]
      4.4

    if -2.474651f-15 < eps < 2.1195103f-10

    1. Started with
      \[\cos \left(x + \varepsilon\right) - \cos x\]
      20.1
    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
      \[\color{red}{\frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)} \leadsto \color{blue}{\left(\varepsilon \cdot \frac{1}{6}\right) \cdot {x}^3 - \varepsilon \cdot \left(\varepsilon \cdot \frac{1}{2} + x\right)}\]
      3.8

    if 2.1195103f-10 < eps

    1. Started with
      \[\cos \left(x + \varepsilon\right) - \cos x\]
      16.7
    2. Using strategy rm
      16.7
    3. Applied cos-sum to get
      \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
      3.0
    4. Applied associate--l- to get
      \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x} \leadsto \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]
      3.0
    5. Using strategy rm
      3.0
    6. Applied flip-- to get
      \[\color{red}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)} \leadsto \color{blue}{\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2}{\cos x \cdot \cos \varepsilon + \left(\sin x \cdot \sin \varepsilon + \cos x\right)}}\]
      3.0
    7. Using strategy rm
      3.0
    8. Applied flip-- to get
      \[\frac{\color{red}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2}}{\cos x \cdot \cos \varepsilon + \left(\sin x \cdot \sin \varepsilon + \cos x\right)} \leadsto \frac{\color{blue}{\frac{{\left({\left(\cos x \cdot \cos \varepsilon\right)}^2\right)}^2 - {\left({\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2\right)}^2}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + {\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2}}}{\cos x \cdot \cos \varepsilon + \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]
      3.0
  1. Removed slow pow expressions

Original test:


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