\[\cos \left(x + \varepsilon\right) - \cos x\]
Test:
NMSE problem 3.3.5
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 17.6 s
Input Error: 17.5
Output Error: 2.6
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{\frac{{\left({\left(\cos \varepsilon \cdot \cos x\right)}^3\right)}^2 - {\left({\left(\cos x + \sin x \cdot \sin \varepsilon\right)}^3\right)}^2}{{\left(\cos \varepsilon \cdot \cos x\right)}^3 + {\left(\cos x + \sin x \cdot \sin \varepsilon\right)}^3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)} & \text{when } \varepsilon \le -6.450162f-09 \\ \left(\varepsilon \cdot \frac{1}{6}\right) \cdot {x}^3 - \varepsilon \cdot \left(\varepsilon \cdot \frac{1}{2} + x\right) & \text{when } \varepsilon \le 0.0018547908f0 \\ \frac{{\left(\cos \varepsilon \cdot \cos x\right)}^{3} - {\left(\cos x + \sin x \cdot \sin \varepsilon\right)}^3}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)} & \text{otherwise} \end{cases}\)

    if eps < -6.450162f-09

    1. Started with
      \[\cos \left(x + \varepsilon\right) - \cos x\]
      15.9
    2. Using strategy rm
      15.9
    3. Applied cos-sum to get
      \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
      2.6
    4. Applied associate--l- to get
      \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x} \leadsto \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]
      2.6
    5. Using strategy rm
      2.6
    6. Applied flip3-- to get
      \[\color{red}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)} \leadsto \color{blue}{\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^{3} - {\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)}}\]
      3.6
    7. Applied simplify to get
      \[\frac{\color{red}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3} - {\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^{3}}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)} \leadsto \frac{\color{blue}{{\left(\cos \varepsilon \cdot \cos x\right)}^3 - {\left(\cos x + \sin x \cdot \sin \varepsilon\right)}^3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)}\]
      2.6
    8. Using strategy rm
      2.6
    9. Applied flip-- to get
      \[\frac{\color{red}{{\left(\cos \varepsilon \cdot \cos x\right)}^3 - {\left(\cos x + \sin x \cdot \sin \varepsilon\right)}^3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)} \leadsto \frac{\color{blue}{\frac{{\left({\left(\cos \varepsilon \cdot \cos x\right)}^3\right)}^2 - {\left({\left(\cos x + \sin x \cdot \sin \varepsilon\right)}^3\right)}^2}{{\left(\cos \varepsilon \cdot \cos x\right)}^3 + {\left(\cos x + \sin x \cdot \sin \varepsilon\right)}^3}}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)}\]
      2.6

    if -6.450162f-09 < eps < 0.0018547908f0

    1. Started with
      \[\cos \left(x + \varepsilon\right) - \cos x\]
      21.6
    2. Applied taylor to get
      \[\cos \left(x + \varepsilon\right) - \cos x \leadsto \frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)\]
      3.7
    3. Taylor expanded around 0 to get
      \[\color{red}{\frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)} \leadsto \color{blue}{\frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)}\]
      3.7
    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 0.0018547908f0 < eps

    1. Started with
      \[\cos \left(x + \varepsilon\right) - \cos x\]
      14.4
    2. Using strategy rm
      14.4
    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. Applied associate--l- to get
      \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x} \leadsto \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]
      0.9
    5. Using strategy rm
      0.9
    6. Applied flip3-- to get
      \[\color{red}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)} \leadsto \color{blue}{\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^{3} - {\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)}}\]
      1.3
    7. Applied simplify to get
      \[\frac{\color{red}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3} - {\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^{3}}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)} \leadsto \frac{\color{blue}{{\left(\cos \varepsilon \cdot \cos x\right)}^3 - {\left(\cos x + \sin x \cdot \sin \varepsilon\right)}^3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)}\]
      0.9
    8. Using strategy rm
      0.9
    9. Applied pow3 to get
      \[\frac{\color{red}{{\left(\cos \varepsilon \cdot \cos x\right)}^3} - {\left(\cos x + \sin x \cdot \sin \varepsilon\right)}^3}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)} \leadsto \frac{\color{blue}{{\left(\cos \varepsilon \cdot \cos x\right)}^{3}} - {\left(\cos x + \sin x \cdot \sin \varepsilon\right)}^3}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)}\]
      1.0
  1. Removed slow pow expressions

Original test:


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