\[\cos \left(x + \varepsilon\right) - \cos x\]
Test:
NMSE problem 3.3.5
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 23.8 s
Input Error: 37.5
Output Error: 5.1
Log: ⚲
Profile: 🕒
\(\begin{cases} \cos \varepsilon \cdot \cos x - (\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_* & \text{when } \varepsilon \le -3.5438160870943954 \cdot 10^{-16} \\ \left({x}^3 \cdot \frac{1}{6} - (\frac{1}{2} * \varepsilon + x)_*\right) \cdot \varepsilon & \text{when } \varepsilon \le 2.1684847443054594 \cdot 10^{-32} \\ \cos \varepsilon \cdot \cos x - (\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_* & \text{otherwise} \end{cases}\)

    if eps < -3.5438160870943954e-16 or 2.1684847443054594e-32 < eps

    1. Started with
      \[\cos \left(x + \varepsilon\right) - \cos x\]
      32.0
    2. Using strategy rm
      32.0
    3. Applied cos-sum to get
      \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
      3.8
    4. Applied associate--l- to get
      \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x} \leadsto \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]
      3.8
    5. Applied simplify to get
      \[\cos x \cdot \cos \varepsilon - \color{red}{\left(\sin x \cdot \sin \varepsilon + \cos x\right)} \leadsto \cos x \cdot \cos \varepsilon - \color{blue}{(\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*}\]
      3.8
    6. Using strategy rm
      3.8
    7. Applied flip-- to get
      \[\color{red}{\cos x \cdot \cos \varepsilon - (\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*} \leadsto \color{blue}{\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left((\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*\right)}^2}{\cos x \cdot \cos \varepsilon + (\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*}}\]
      4.2
    8. Applied simplify to get
      \[\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left((\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*\right)}^2}{\color{red}{\cos x \cdot \cos \varepsilon + (\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*}} \leadsto \frac{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left((\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*\right)}^2}{\color{blue}{(\left(\cos \varepsilon\right) * \left(\cos x\right) + \left((\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*\right))_*}}\]
      4.2
    9. Applied taylor to get
      \[\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left((\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*\right)}^2}{(\left(\cos \varepsilon\right) * \left(\cos x\right) + \left((\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*\right))_*} \leadsto \frac{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left((\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*\right)}^2}{(\left(\cos \varepsilon\right) * \left(\cos x\right) + \left((\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*\right))_*}\]
      4.2
    10. Taylor expanded around 0 to get
      \[\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left((\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*\right)}^2}{\color{red}{(\left(\cos \varepsilon\right) * \left(\cos x\right) + \left((\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*\right))_*}} \leadsto \frac{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left((\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*\right)}^2}{\color{blue}{(\left(\cos \varepsilon\right) * \left(\cos x\right) + \left((\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*\right))_*}}\]
      4.2
    11. Applied simplify to get
      \[\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left((\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*\right)}^2}{(\left(\cos \varepsilon\right) * \left(\cos x\right) + \left((\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*\right))_*} \leadsto \frac{(\left(\cos x\right) * \left(\cos \varepsilon\right) + \left((\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*\right))_*}{\frac{(\left(\cos x\right) * \left(\cos \varepsilon\right) + \left((\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*\right))_*}{\cos \varepsilon \cdot \cos x - (\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*}}\]
      3.8
    12. Applied final simplification
    13. Applied simplify to get
      \[\color{red}{\frac{(\left(\cos x\right) * \left(\cos \varepsilon\right) + \left((\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*\right))_*}{\frac{(\left(\cos x\right) * \left(\cos \varepsilon\right) + \left((\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*\right))_*}{\cos \varepsilon \cdot \cos x - (\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*}}} \leadsto \color{blue}{\cos \varepsilon \cdot \cos x - (\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*}\]
      3.8

    if -3.5438160870943954e-16 < eps < 2.1684847443054594e-32

    1. Started with
      \[\cos \left(x + \varepsilon\right) - \cos x\]
      45.9
    2. Applied taylor to get
      \[\cos \left(x + \varepsilon\right) - \cos x \leadsto \frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)\]
      7.1
    3. Taylor expanded around 0 to get
      \[\color{red}{\frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)} \leadsto \color{blue}{\frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)}\]
      7.1
    4. Applied simplify to get
      \[\color{red}{\frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)} \leadsto \color{blue}{\left(\varepsilon \cdot \frac{1}{6}\right) \cdot {x}^3 - \varepsilon \cdot (\frac{1}{2} * \varepsilon + x)_*}\]
      7.2
    5. Applied taylor to get
      \[\left(\varepsilon \cdot \frac{1}{6}\right) \cdot {x}^3 - \varepsilon \cdot (\frac{1}{2} * \varepsilon + x)_* \leadsto \varepsilon \cdot \left(\frac{1}{6} \cdot {x}^3 - (\frac{1}{2} * \varepsilon + x)_*\right)\]
      7.2
    6. Taylor expanded around 0 to get
      \[\color{red}{\varepsilon \cdot \left(\frac{1}{6} \cdot {x}^3 - (\frac{1}{2} * \varepsilon + x)_*\right)} \leadsto \color{blue}{\varepsilon \cdot \left(\frac{1}{6} \cdot {x}^3 - (\frac{1}{2} * \varepsilon + x)_*\right)}\]
      7.2
    7. Applied simplify to get
      \[\varepsilon \cdot \left(\frac{1}{6} \cdot {x}^3 - (\frac{1}{2} * \varepsilon + x)_*\right) \leadsto \left({x}^3 \cdot \frac{1}{6} - (\frac{1}{2} * \varepsilon + x)_*\right) \cdot \varepsilon\]
      7.2
    8. 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)))