\[\cos \left(x + \varepsilon\right) - \cos x\]
Test:
NMSE problem 3.3.5
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 28.3 s
Input Error: 39.7
Output Error: 24.5
Log: ⚲
Profile: 🕒
\(\left(\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^3}{(\left(\sin x \cdot \sin \varepsilon\right) * \left(\cos x \cdot \cos \varepsilon\right) + \left({\left(\cos x \cdot \cos \varepsilon\right)}^2\right))_* + {\left(\sin x \cdot \sin \varepsilon\right)}^2} - \cos x\right) - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^3}{(\left(\sin x \cdot \sin \varepsilon\right) * \left(\cos x \cdot \cos \varepsilon\right) + \left({\left(\cos x \cdot \cos \varepsilon\right)}^2\right))_* + {\left(\sin x \cdot \sin \varepsilon\right)}^2}\)
  1. Started with
    \[\cos \left(x + \varepsilon\right) - \cos x\]
    39.7
  2. Using strategy rm
    39.7
  3. Applied cos-sum to get
    \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
    24.5
  4. Using strategy rm
    24.5
  5. Applied flip3-- to get
    \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x \leadsto \color{blue}{\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^{3} - {\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)}} - \cos x\]
    24.6
  6. Applied simplify to get
    \[\frac{\color{red}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3} - {\left(\sin x \cdot \sin \varepsilon\right)}^{3}}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)} - \cos x \leadsto \frac{\color{blue}{{\left(\cos \varepsilon \cdot \cos x\right)}^3 - {\left(\sin x \cdot \sin \varepsilon\right)}^3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)} - \cos x\]
    24.6
  7. Using strategy rm
    24.6
  8. Applied add-cbrt-cube to get
    \[\color{red}{\frac{{\left(\cos \varepsilon \cdot \cos x\right)}^3 - {\left(\sin x \cdot \sin \varepsilon\right)}^3}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)} - \cos x} \leadsto \color{blue}{\sqrt[3]{{\left(\frac{{\left(\cos \varepsilon \cdot \cos x\right)}^3 - {\left(\sin x \cdot \sin \varepsilon\right)}^3}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)} - \cos x\right)}^3}}\]
    24.7
  9. Applied taylor to get
    \[\sqrt[3]{{\left(\frac{{\left(\cos \varepsilon \cdot \cos x\right)}^3 - {\left(\sin x \cdot \sin \varepsilon\right)}^3}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)} - \cos x\right)}^3} \leadsto \sqrt[3]{{\left(\frac{{\left(\cos \varepsilon \cdot \cos x\right)}^3}{{\left(\cos x\right)}^2 \cdot {\left(\cos \varepsilon\right)}^2 + \left(\sin x \cdot \left(\sin \varepsilon \cdot \left(\cos x \cdot \cos \varepsilon\right)\right) + {\left(\sin x\right)}^2 \cdot {\left(\sin \varepsilon\right)}^2\right)} - \left(\cos x + \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^3}{{\left(\cos x\right)}^2 \cdot {\left(\cos \varepsilon\right)}^2 + \left(\sin x \cdot \left(\sin \varepsilon \cdot \left(\cos x \cdot \cos \varepsilon\right)\right) + {\left(\sin x\right)}^2 \cdot {\left(\sin \varepsilon\right)}^2\right)}\right)\right)}^3}\]
    24.7
  10. Taylor expanded around 0 to get
    \[\sqrt[3]{\color{red}{{\left(\frac{{\left(\cos \varepsilon \cdot \cos x\right)}^3}{{\left(\cos x\right)}^2 \cdot {\left(\cos \varepsilon\right)}^2 + \left(\sin x \cdot \left(\sin \varepsilon \cdot \left(\cos x \cdot \cos \varepsilon\right)\right) + {\left(\sin x\right)}^2 \cdot {\left(\sin \varepsilon\right)}^2\right)} - \left(\cos x + \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^3}{{\left(\cos x\right)}^2 \cdot {\left(\cos \varepsilon\right)}^2 + \left(\sin x \cdot \left(\sin \varepsilon \cdot \left(\cos x \cdot \cos \varepsilon\right)\right) + {\left(\sin x\right)}^2 \cdot {\left(\sin \varepsilon\right)}^2\right)}\right)\right)}^3}} \leadsto \sqrt[3]{\color{blue}{{\left(\frac{{\left(\cos \varepsilon \cdot \cos x\right)}^3}{{\left(\cos x\right)}^2 \cdot {\left(\cos \varepsilon\right)}^2 + \left(\sin x \cdot \left(\sin \varepsilon \cdot \left(\cos x \cdot \cos \varepsilon\right)\right) + {\left(\sin x\right)}^2 \cdot {\left(\sin \varepsilon\right)}^2\right)} - \left(\cos x + \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^3}{{\left(\cos x\right)}^2 \cdot {\left(\cos \varepsilon\right)}^2 + \left(\sin x \cdot \left(\sin \varepsilon \cdot \left(\cos x \cdot \cos \varepsilon\right)\right) + {\left(\sin x\right)}^2 \cdot {\left(\sin \varepsilon\right)}^2\right)}\right)\right)}^3}}\]
    24.7
  11. Applied simplify to get
    \[\sqrt[3]{{\left(\frac{{\left(\cos \varepsilon \cdot \cos x\right)}^3}{{\left(\cos x\right)}^2 \cdot {\left(\cos \varepsilon\right)}^2 + \left(\sin x \cdot \left(\sin \varepsilon \cdot \left(\cos x \cdot \cos \varepsilon\right)\right) + {\left(\sin x\right)}^2 \cdot {\left(\sin \varepsilon\right)}^2\right)} - \left(\cos x + \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^3}{{\left(\cos x\right)}^2 \cdot {\left(\cos \varepsilon\right)}^2 + \left(\sin x \cdot \left(\sin \varepsilon \cdot \left(\cos x \cdot \cos \varepsilon\right)\right) + {\left(\sin x\right)}^2 \cdot {\left(\sin \varepsilon\right)}^2\right)}\right)\right)}^3} \leadsto \left(\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^3}{(\left(\sin x \cdot \sin \varepsilon\right) * \left(\cos x \cdot \cos \varepsilon\right) + \left({\left(\cos x \cdot \cos \varepsilon\right)}^2\right))_* + {\left(\sin x \cdot \sin \varepsilon\right)}^2} - \cos x\right) - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^3}{(\left(\sin x \cdot \sin \varepsilon\right) * \left(\cos x \cdot \cos \varepsilon\right) + \left({\left(\cos x \cdot \cos \varepsilon\right)}^2\right))_* + {\left(\sin x \cdot \sin \varepsilon\right)}^2}\]
    24.5
  12. Applied final simplification

Original test:


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