\[\cos \left(x + \varepsilon\right) - \cos x\]
Test:
NMSE problem 3.3.5
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 1.1 m
Input Error: 37.1
Output Error: 3.8
Log: ⚲
\(\begin{cases} \frac{{\left(\cos \varepsilon \cdot \cos x\right)}^3 - {\left(\sin \varepsilon \cdot \sin x + \cos x\right)}^3}{{\left(\cos \varepsilon \cdot \cos x\right)}^2 + \left(\cos \varepsilon \cdot \cos x + \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right) \cdot \left(\sin x \cdot \sin \varepsilon + \cos x\right)} & \text{when } \varepsilon \le -1.3296779497386022 \cdot 10^{-08} \\ \left(\varepsilon \cdot \frac{1}{6}\right) \cdot {x}^3 - \varepsilon \cdot \left(\frac{1}{2} \cdot \varepsilon + x\right) & \text{when } \varepsilon \le 7.134416671297405 \cdot 10^{-12} \\ \frac{{\left(\cos \varepsilon \cdot \cos x\right)}^3 - {\left(\sin \varepsilon \cdot \sin x + \cos x\right)}^3}{{\left(\cos \varepsilon \cdot \cos x\right)}^2 + \left(\cos \varepsilon \cdot \cos x + \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right) \cdot \left(\sin x \cdot \sin \varepsilon + \cos x\right)} & \text{otherwise} \end{cases}\)

    if eps < -1.3296779497386022e-08

    1. Started with
      \[\cos \left(x + \varepsilon\right) - \cos x\]
      30.7
    2. Using strategy rm
      30.7
    3. Applied cos-sum to get
      \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
      1.1
    4. Applied associate--l- to get
      \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x} \leadsto \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]
      1.1
    5. Using strategy rm
      1.1
    6. Applied flip3-- to get
      \[\color{red}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)} \leadsto \color{blue}{\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^{3} - {\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)}}\]
      1.3
    7. Applied simplify to get
      \[\frac{\color{red}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3} - {\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^{3}}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)} \leadsto \frac{\color{blue}{{\left(\cos \varepsilon \cdot \cos x\right)}^3 - {\left(\sin \varepsilon \cdot \sin x + \cos x\right)}^3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)}\]
      1.3
    8. Applied simplify to get
      \[\frac{{\left(\cos \varepsilon \cdot \cos x\right)}^3 - {\left(\sin \varepsilon \cdot \sin x + \cos x\right)}^3}{\color{red}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)}} \leadsto \frac{{\left(\cos \varepsilon \cdot \cos x\right)}^3 - {\left(\sin \varepsilon \cdot \sin x + \cos x\right)}^3}{\color{blue}{{\left(\cos \varepsilon \cdot \cos x\right)}^2 + \left(\cos \varepsilon \cdot \cos x + \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right) \cdot \left(\sin x \cdot \sin \varepsilon + \cos x\right)}}\]
      1.3

    if -1.3296779497386022e-08 < eps < 7.134416671297405e-12

    1. Started with
      \[\cos \left(x + \varepsilon\right) - \cos x\]
      46.0
    2. Applied taylor to get
      \[\cos \left(x + \varepsilon\right) - \cos x \leadsto \frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)\]
      6.9
    3. Taylor expanded around 0 to get
      \[\color{red}{\frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)} \leadsto \color{blue}{\frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)}\]
      6.9
    4. Applied simplify to get
      \[\color{red}{\frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)} \leadsto \color{blue}{\left(\varepsilon \cdot \frac{1}{6}\right) \cdot {x}^3 - \varepsilon \cdot \left(\frac{1}{2} \cdot \varepsilon + x\right)}\]
      7.0

    if 7.134416671297405e-12 < eps

    1. Started with
      \[\cos \left(x + \varepsilon\right) - \cos x\]
      30.4
    2. Using strategy rm
      30.4
    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
      \[\color{red}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)} \leadsto \color{blue}{\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^{3} - {\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)}}\]
      1.8
    7. Applied simplify to get
      \[\frac{\color{red}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3} - {\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^{3}}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)} \leadsto \frac{\color{blue}{{\left(\cos \varepsilon \cdot \cos x\right)}^3 - {\left(\sin \varepsilon \cdot \sin x + \cos x\right)}^3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)}\]
      1.8
    8. Applied simplify to get
      \[\frac{{\left(\cos \varepsilon \cdot \cos x\right)}^3 - {\left(\sin \varepsilon \cdot \sin x + \cos x\right)}^3}{\color{red}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)}} \leadsto \frac{{\left(\cos \varepsilon \cdot \cos x\right)}^3 - {\left(\sin \varepsilon \cdot \sin x + \cos x\right)}^3}{\color{blue}{{\left(\cos \varepsilon \cdot \cos x\right)}^2 + \left(\cos \varepsilon \cdot \cos x + \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right) \cdot \left(\sin x \cdot \sin \varepsilon + \cos x\right)}}\]
      1.8
  1. Removed slow pow expressions

Original test:


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