\[\cos \left(x + \varepsilon\right) - \cos x\]
Test:
NMSE problem 3.3.5
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 15.5 s
Input Error: 18.3
Output Error: 0.6
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{{\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right)}^3 - {\left(\cos x\right)}^3}{(\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) * \left((\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin \varepsilon \cdot \sin x\right) + \left(\cos x \cdot \cos x\right))_*} & \text{when } \varepsilon \le -0.00030713418f0 \\ \frac{1}{6} \cdot \left({\varepsilon}^3 \cdot \sin x\right) - (\left(\frac{1}{2} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) * \left(\cos x\right) + \left(\sin x \cdot \varepsilon\right))_* & \text{when } \varepsilon \le 0.07828799f0 \\ \frac{{\left(\cos x \cdot \cos \varepsilon\right)}^3 - {\left(\sin x \cdot \sin \varepsilon\right)}^3}{(\left(\sin x \cdot \sin \varepsilon\right) * \left((\left(\sin x\right) * \left(\sin \varepsilon\right) + \left(\cos x \cdot \cos \varepsilon\right))_*\right) + \left({\left(\cos x \cdot \cos \varepsilon\right)}^2\right))_*} - \cos x & \text{otherwise} \end{cases}\)

    if eps < -0.00030713418f0

    1. Started with
      \[\cos \left(x + \varepsilon\right) - \cos x\]
      14.6
    2. Using strategy rm
      14.6
    3. Applied cos-sum to get
      \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
      1.2
    4. Using strategy rm
      1.2
    5. Applied flip3-- to get
      \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x} \leadsto \color{blue}{\frac{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^{3} - {\left(\cos x\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^2 + \left({\left(\cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) \cdot \cos x\right)}}\]
      1.3
    6. Applied simplify to get
      \[\frac{\color{red}{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^{3} - {\left(\cos x\right)}^{3}}}{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^2 + \left({\left(\cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) \cdot \cos x\right)} \leadsto \frac{\color{blue}{{\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right)}^3 - {\left(\cos x\right)}^3}}{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^2 + \left({\left(\cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) \cdot \cos x\right)}\]
      1.2
    7. Applied simplify to get
      \[\frac{{\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right)}^3 - {\left(\cos x\right)}^3}{\color{red}{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^2 + \left({\left(\cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) \cdot \cos x\right)}} \leadsto \frac{{\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right)}^3 - {\left(\cos x\right)}^3}{\color{blue}{(\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) * \left((\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin \varepsilon \cdot \sin x\right) + \left(\cos x \cdot \cos x\right))_*}}\]
      1.3

    if -0.00030713418f0 < eps < 0.07828799f0

    1. Started with
      \[\cos \left(x + \varepsilon\right) - \cos x\]
      22.8
    2. Using strategy rm
      22.8
    3. Applied add-cube-cbrt to get
      \[\color{red}{\cos \left(x + \varepsilon\right) - \cos x} \leadsto \color{blue}{{\left(\sqrt[3]{\cos \left(x + \varepsilon\right) - \cos x}\right)}^3}\]
      22.8
    4. Applied taylor to get
      \[{\left(\sqrt[3]{\cos \left(x + \varepsilon\right) - \cos x}\right)}^3 \leadsto \frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot \sin x\right) - \left(\frac{1}{2} \cdot \left({\varepsilon}^2 \cdot \cos x\right) + \varepsilon \cdot \sin x\right)\]
      0.1
    5. Taylor expanded around 0 to get
      \[\color{red}{\frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot \sin x\right) - \left(\frac{1}{2} \cdot \left({\varepsilon}^2 \cdot \cos x\right) + \varepsilon \cdot \sin x\right)} \leadsto \color{blue}{\frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot \sin x\right) - \left(\frac{1}{2} \cdot \left({\varepsilon}^2 \cdot \cos x\right) + \varepsilon \cdot \sin x\right)}\]
      0.1
    6. Applied simplify to get
      \[\frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot \sin x\right) - \left(\frac{1}{2} \cdot \left({\varepsilon}^2 \cdot \cos x\right) + \varepsilon \cdot \sin x\right) \leadsto \frac{1}{6} \cdot \left({\varepsilon}^3 \cdot \sin x\right) - (\left(\frac{1}{2} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) * \left(\cos x\right) + \left(\sin x \cdot \varepsilon\right))_*\]
      0.2
    7. Applied final simplification

    if 0.07828799f0 < eps

    1. Started with
      \[\cos \left(x + \varepsilon\right) - \cos x\]
      13.9
    2. Using strategy rm
      13.9
    3. Applied cos-sum to get
      \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
      0.7
    4. Using strategy rm
      0.7
    5. Applied flip3-- to get
      \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x \leadsto \color{blue}{\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^{3} - {\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)}} - \cos x\]
      1.0
    6. Applied simplify to get
      \[\frac{\color{red}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3} - {\left(\sin x \cdot \sin \varepsilon\right)}^{3}}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)} - \cos x \leadsto \frac{\color{blue}{{\left(\cos \varepsilon \cdot \cos x\right)}^3 - {\left(\sin x \cdot \sin \varepsilon\right)}^3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)} - \cos x\]
      0.7
    7. Applied taylor to get
      \[\frac{{\left(\cos \varepsilon \cdot \cos x\right)}^3 - {\left(\sin x \cdot \sin \varepsilon\right)}^3}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)} - \cos x \leadsto \frac{{\left(\cos \varepsilon \cdot \cos x\right)}^3 - {\left(\sin x \cdot \sin \varepsilon\right)}^3}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)} - \cos x\]
      0.7
    8. Taylor expanded around 0 to get
      \[\frac{\color{red}{{\left(\cos \varepsilon \cdot \cos x\right)}^3} - {\left(\sin x \cdot \sin \varepsilon\right)}^3}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)} - \cos x \leadsto \frac{\color{blue}{{\left(\cos \varepsilon \cdot \cos x\right)}^3} - {\left(\sin x \cdot \sin \varepsilon\right)}^3}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)} - \cos x\]
      0.7
    9. Applied simplify to get
      \[\frac{{\left(\cos \varepsilon \cdot \cos x\right)}^3 - {\left(\sin x \cdot \sin \varepsilon\right)}^3}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)} - \cos x \leadsto \frac{{\left(\cos x \cdot \cos \varepsilon\right)}^3 - {\left(\sin x \cdot \sin \varepsilon\right)}^3}{(\left(\sin x \cdot \sin \varepsilon\right) * \left((\left(\sin x\right) * \left(\sin \varepsilon\right) + \left(\cos x \cdot \cos \varepsilon\right))_*\right) + \left({\left(\cos x \cdot \cos \varepsilon\right)}^2\right))_*} - \cos x\]
      0.8
    10. Applied final simplification
  1. Removed slow pow expressions

Original test:


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