\[\cos \left(x + \varepsilon\right) - \cos x\]
Test:
NMSE problem 3.3.5
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 21.4 s
Input Error: 18.6
Output Error: 0.6
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{{\left({\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2\right)}^{1}}{\cos x \cdot \cos \varepsilon + \left(\sin x \cdot \sin \varepsilon + \cos x\right)} & \text{when } \varepsilon \le -0.013343895f0 \\ \left(\sin x \cdot \left(\varepsilon \cdot \frac{1}{6}\right) - \frac{1}{2} \cdot \cos x\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \varepsilon \cdot \sin x & \text{when } \varepsilon \le 0.08563255f0 \\ \log \left(e^{\left(\cos \varepsilon \cdot \cos x - \cos x\right) - \sin x \cdot \sin \varepsilon}\right) & \text{otherwise} \end{cases}\)

    if eps < -0.013343895f0

    1. Started with
      \[\cos \left(x + \varepsilon\right) - \cos x\]
      14.7
    2. Using strategy rm
      14.7
    3. Applied cos-sum to get
      \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
      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 flip-- to get
      \[\color{red}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)} \leadsto \color{blue}{\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2}{\cos x \cdot \cos \varepsilon + \left(\sin x \cdot \sin \varepsilon + \cos x\right)}}\]
      0.9
    7. Using strategy rm
      0.9
    8. Applied pow1 to get
      \[\frac{\color{red}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2}}{\cos x \cdot \cos \varepsilon + \left(\sin x \cdot \sin \varepsilon + \cos x\right)} \leadsto \frac{\color{blue}{{\left({\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2\right)}^{1}}}{\cos x \cdot \cos \varepsilon + \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]
      0.9

    if -0.013343895f0 < eps < 0.08563255f0

    1. Started with
      \[\cos \left(x + \varepsilon\right) - \cos x\]
      23.0
    2. Using strategy rm
      23.0
    3. Applied cos-sum to get
      \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
      18.3
    4. Applied associate--l- to get
      \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x} \leadsto \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]
      18.3
    5. Using strategy rm
      18.3
    6. Applied add-cube-cbrt to get
      \[\color{red}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)} \leadsto \color{blue}{{\left(\sqrt[3]{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\right)}^3}\]
      18.3
    7. Applied taylor to get
      \[{\left(\sqrt[3]{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\right)}^3 \leadsto \frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot \sin x\right) - \left(\frac{1}{2} \cdot \left({\varepsilon}^2 \cdot \cos x\right) + \varepsilon \cdot \sin x\right)\]
      0.2
    8. Taylor expanded around 0 to get
      \[\color{red}{\frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot \sin x\right) - \left(\frac{1}{2} \cdot \left({\varepsilon}^2 \cdot \cos x\right) + \varepsilon \cdot \sin x\right)} \leadsto \color{blue}{\frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot \sin x\right) - \left(\frac{1}{2} \cdot \left({\varepsilon}^2 \cdot \cos x\right) + \varepsilon \cdot \sin x\right)}\]
      0.2
    9. Applied simplify to get
      \[\frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot \sin x\right) - \left(\frac{1}{2} \cdot \left({\varepsilon}^2 \cdot \cos x\right) + \varepsilon \cdot \sin x\right) \leadsto \left(\frac{1}{6} \cdot \left({\varepsilon}^3 \cdot \sin x\right) - \frac{1}{2} \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \cos x\right)\right) - \sin x \cdot \varepsilon\]
      0.2
    10. Applied final simplification
    11. Applied simplify to get
      \[\color{red}{\left(\frac{1}{6} \cdot \left({\varepsilon}^3 \cdot \sin x\right) - \frac{1}{2} \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \cos x\right)\right) - \sin x \cdot \varepsilon} \leadsto \color{blue}{\left(\sin x \cdot \left(\varepsilon \cdot \frac{1}{6}\right) - \frac{1}{2} \cdot \cos x\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \varepsilon \cdot \sin x}\]
      0.2

    if 0.08563255f0 < eps

    1. Started with
      \[\cos \left(x + \varepsilon\right) - \cos x\]
      14.3
    2. Using strategy rm
      14.3
    3. Applied cos-sum to get
      \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
      0.7
    4. Applied associate--l- to get
      \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x} \leadsto \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]
      0.7
    5. Using strategy rm
      0.7
    6. Applied add-log-exp to get
      \[\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \color{red}{\cos x}\right) \leadsto \cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \color{blue}{\log \left(e^{\cos x}\right)}\right)\]
      0.9
    7. Applied add-log-exp to get
      \[\cos x \cdot \cos \varepsilon - \left(\color{red}{\sin x \cdot \sin \varepsilon} + \log \left(e^{\cos x}\right)\right) \leadsto \cos x \cdot \cos \varepsilon - \left(\color{blue}{\log \left(e^{\sin x \cdot \sin \varepsilon}\right)} + \log \left(e^{\cos x}\right)\right)\]
      1.0
    8. Applied sum-log to get
      \[\cos x \cdot \cos \varepsilon - \color{red}{\left(\log \left(e^{\sin x \cdot \sin \varepsilon}\right) + \log \left(e^{\cos x}\right)\right)} \leadsto \cos x \cdot \cos \varepsilon - \color{blue}{\log \left(e^{\sin x \cdot \sin \varepsilon} \cdot e^{\cos x}\right)}\]
      1.0
    9. Applied add-log-exp to get
      \[\color{red}{\cos x \cdot \cos \varepsilon} - \log \left(e^{\sin x \cdot \sin \varepsilon} \cdot e^{\cos x}\right) \leadsto \color{blue}{\log \left(e^{\cos x \cdot \cos \varepsilon}\right)} - \log \left(e^{\sin x \cdot \sin \varepsilon} \cdot e^{\cos x}\right)\]
      1.2
    10. Applied diff-log to get
      \[\color{red}{\log \left(e^{\cos x \cdot \cos \varepsilon}\right) - \log \left(e^{\sin x \cdot \sin \varepsilon} \cdot e^{\cos x}\right)} \leadsto \color{blue}{\log \left(\frac{e^{\cos x \cdot \cos \varepsilon}}{e^{\sin x \cdot \sin \varepsilon} \cdot e^{\cos x}}\right)}\]
      1.2
    11. Applied simplify to get
      \[\log \color{red}{\left(\frac{e^{\cos x \cdot \cos \varepsilon}}{e^{\sin x \cdot \sin \varepsilon} \cdot e^{\cos x}}\right)} \leadsto \log \color{blue}{\left(e^{\left(\cos \varepsilon \cdot \cos x - \cos x\right) - \sin x \cdot \sin \varepsilon}\right)}\]
      0.9
  1. Removed slow pow expressions

Original test:


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