\[\cos \left(x + \varepsilon\right) - \cos x\]
Test:
NMSE problem 3.3.5
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 18.0 s
Input Error: 17.5
Output Error: 3.2
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{{\left(\cos \varepsilon \cdot \cos x\right)}^{3} - {\left(\cos x + \sin x \cdot \sin \varepsilon\right)}^3}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)} & \text{when } \varepsilon \le -1.9234536f-05 \\ \left(\varepsilon \cdot \frac{1}{6}\right) \cdot {x}^3 - \varepsilon \cdot \left(\varepsilon \cdot \frac{1}{2} + x\right) & \text{when } \varepsilon \le 5.083976f-14 \\ \frac{\frac{{\left({\left(\cos x \cdot \cos \varepsilon\right)}^3\right)}^3 - {\left({\left(\cos x + \sin x \cdot \sin \varepsilon\right)}^3\right)}^3}{{\left(\cos x \cdot \cos \varepsilon\right)}^3 \cdot \left({\left(\cos x + \sin x \cdot \sin \varepsilon\right)}^3 + {\left(\cos x \cdot \cos \varepsilon\right)}^3\right) + {\left({\left(\cos x + \sin x \cdot \sin \varepsilon\right)}^3\right)}^2}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)} & \text{otherwise} \end{cases}\)

    if eps < -1.9234536f-05

    1. Started with
      \[\cos \left(x + \varepsilon\right) - \cos x\]
      15.1
    2. Using strategy rm
      15.1
    3. Applied cos-sum to get
      \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
      1.6
    4. Applied associate--l- to get
      \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x} \leadsto \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]
      1.6
    5. Using strategy rm
      1.6
    6. Applied flip3-- to get
      \[\color{red}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)} \leadsto \color{blue}{\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^{3} - {\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)}}\]
      2.0
    7. Applied simplify to get
      \[\frac{\color{red}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3} - {\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^{3}}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)} \leadsto \frac{\color{blue}{{\left(\cos \varepsilon \cdot \cos x\right)}^3 - {\left(\cos x + \sin x \cdot \sin \varepsilon\right)}^3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)}\]
      1.6
    8. Using strategy rm
      1.6
    9. Applied pow3 to get
      \[\frac{\color{red}{{\left(\cos \varepsilon \cdot \cos x\right)}^3} - {\left(\cos x + \sin x \cdot \sin \varepsilon\right)}^3}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)} \leadsto \frac{\color{blue}{{\left(\cos \varepsilon \cdot \cos x\right)}^{3}} - {\left(\cos x + \sin x \cdot \sin \varepsilon\right)}^3}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)}\]
      1.7

    if -1.9234536f-05 < eps < 5.083976f-14

    1. Started with
      \[\cos \left(x + \varepsilon\right) - \cos x\]
      20.7
    2. Applied taylor to get
      \[\cos \left(x + \varepsilon\right) - \cos x \leadsto \frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)\]
      4.0
    3. Taylor expanded around 0 to get
      \[\color{red}{\frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)} \leadsto \color{blue}{\frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)}\]
      4.0
    4. Applied simplify to get
      \[\color{red}{\frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)} \leadsto \color{blue}{\left(\varepsilon \cdot \frac{1}{6}\right) \cdot {x}^3 - \varepsilon \cdot \left(\varepsilon \cdot \frac{1}{2} + x\right)}\]
      4.0

    if 5.083976f-14 < eps

    1. Started with
      \[\cos \left(x + \varepsilon\right) - \cos x\]
      16.8
    2. Using strategy rm
      16.8
    3. Applied cos-sum to get
      \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
      3.6
    4. Applied associate--l- to get
      \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x} \leadsto \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]
      3.6
    5. Using strategy rm
      3.6
    6. Applied flip3-- to get
      \[\color{red}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)} \leadsto \color{blue}{\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^{3} - {\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)}}\]
      5.8
    7. Applied simplify to get
      \[\frac{\color{red}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3} - {\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^{3}}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)} \leadsto \frac{\color{blue}{{\left(\cos \varepsilon \cdot \cos x\right)}^3 - {\left(\cos x + \sin x \cdot \sin \varepsilon\right)}^3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)}\]
      3.6
    8. Using strategy rm
      3.6
    9. Applied flip3-- to get
      \[\frac{\color{red}{{\left(\cos \varepsilon \cdot \cos x\right)}^3 - {\left(\cos x + \sin x \cdot \sin \varepsilon\right)}^3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)} \leadsto \frac{\color{blue}{\frac{{\left({\left(\cos \varepsilon \cdot \cos x\right)}^3\right)}^{3} - {\left({\left(\cos x + \sin x \cdot \sin \varepsilon\right)}^3\right)}^{3}}{{\left({\left(\cos \varepsilon \cdot \cos x\right)}^3\right)}^2 + \left({\left({\left(\cos x + \sin x \cdot \sin \varepsilon\right)}^3\right)}^2 + {\left(\cos \varepsilon \cdot \cos x\right)}^3 \cdot {\left(\cos x + \sin x \cdot \sin \varepsilon\right)}^3\right)}}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)}\]
      5.7
    10. Applied simplify to get
      \[\frac{\frac{\color{red}{{\left({\left(\cos \varepsilon \cdot \cos x\right)}^3\right)}^{3} - {\left({\left(\cos x + \sin x \cdot \sin \varepsilon\right)}^3\right)}^{3}}}{{\left({\left(\cos \varepsilon \cdot \cos x\right)}^3\right)}^2 + \left({\left({\left(\cos x + \sin x \cdot \sin \varepsilon\right)}^3\right)}^2 + {\left(\cos \varepsilon \cdot \cos x\right)}^3 \cdot {\left(\cos x + \sin x \cdot \sin \varepsilon\right)}^3\right)}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)} \leadsto \frac{\frac{\color{blue}{{\left({\left(\cos x \cdot \cos \varepsilon\right)}^3\right)}^3 - {\left({\left(\cos x + \sin x \cdot \sin \varepsilon\right)}^3\right)}^3}}{{\left({\left(\cos \varepsilon \cdot \cos x\right)}^3\right)}^2 + \left({\left({\left(\cos x + \sin x \cdot \sin \varepsilon\right)}^3\right)}^2 + {\left(\cos \varepsilon \cdot \cos x\right)}^3 \cdot {\left(\cos x + \sin x \cdot \sin \varepsilon\right)}^3\right)}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)}\]
      3.6
    11. Applied simplify to get
      \[\frac{\frac{{\left({\left(\cos x \cdot \cos \varepsilon\right)}^3\right)}^3 - {\left({\left(\cos x + \sin x \cdot \sin \varepsilon\right)}^3\right)}^3}{\color{red}{{\left({\left(\cos \varepsilon \cdot \cos x\right)}^3\right)}^2 + \left({\left({\left(\cos x + \sin x \cdot \sin \varepsilon\right)}^3\right)}^2 + {\left(\cos \varepsilon \cdot \cos x\right)}^3 \cdot {\left(\cos x + \sin x \cdot \sin \varepsilon\right)}^3\right)}}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)} \leadsto \frac{\frac{{\left({\left(\cos x \cdot \cos \varepsilon\right)}^3\right)}^3 - {\left({\left(\cos x + \sin x \cdot \sin \varepsilon\right)}^3\right)}^3}{\color{blue}{{\left(\cos x \cdot \cos \varepsilon\right)}^3 \cdot \left({\left(\cos x + \sin x \cdot \sin \varepsilon\right)}^3 + {\left(\cos x \cdot \cos \varepsilon\right)}^3\right) + {\left({\left(\cos x + \sin x \cdot \sin \varepsilon\right)}^3\right)}^2}}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)}\]
      3.6
  1. Removed slow pow expressions

Original test:


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