\[\cos \left(x + \varepsilon\right) - \cos x\]
Test:
NMSE problem 3.3.5
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 29.1 s
Input Error: 37.2
Output Error: 3.8
Log: ⚲
Profile: 🕒
\(\begin{cases} \log_* (1 + (e^{\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon} - 1)^*) - \cos x & \text{when } \varepsilon \le -6.974478371048821 \cdot 10^{-14} \\ \left(\varepsilon \cdot \frac{1}{6}\right) \cdot {x}^3 - \varepsilon \cdot (\frac{1}{2} * \varepsilon + x)_* & \text{when } \varepsilon \le 0.00018005451351709028 \\ \frac{{\left(\cos \varepsilon \cdot \cos x\right)}^3 - {\left(\sin x \cdot \sin \varepsilon\right)}^3}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)} - \cos x & \text{otherwise} \end{cases}\)

    if eps < -6.974478371048821e-14

    1. Started with
      \[\cos \left(x + \varepsilon\right) - \cos x\]
      30.9
    2. Using strategy rm
      30.9
    3. Applied cos-sum to get
      \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
      1.9
    4. Using strategy rm
      1.9
    5. Applied log1p-expm1-u to get
      \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x \leadsto \color{blue}{\log_* (1 + (e^{\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon} - 1)^*)} - \cos x\]
      2.0

    if -6.974478371048821e-14 < eps < 0.00018005451351709028

    1. Started with
      \[\cos \left(x + \varepsilon\right) - \cos x\]
      46.0
    2. Applied taylor to get
      \[\cos \left(x + \varepsilon\right) - \cos x \leadsto \frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)\]
      6.8
    3. Taylor expanded around 0 to get
      \[\color{red}{\frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)} \leadsto \color{blue}{\frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)}\]
      6.8
    4. Applied simplify to get
      \[\color{red}{\frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)} \leadsto \color{blue}{\left(\varepsilon \cdot \frac{1}{6}\right) \cdot {x}^3 - \varepsilon \cdot (\frac{1}{2} * \varepsilon + x)_*}\]
      6.8

    if 0.00018005451351709028 < eps

    1. Started with
      \[\cos \left(x + \varepsilon\right) - \cos x\]
      30.7
    2. Using strategy rm
      30.7
    3. Applied cos-sum to get
      \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
      0.9
    4. Using strategy rm
      0.9
    5. Applied flip3-- to get
      \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x \leadsto \color{blue}{\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^{3} - {\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)}} - \cos x\]
      1.1
    6. Applied simplify to get
      \[\frac{\color{red}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3} - {\left(\sin x \cdot \sin \varepsilon\right)}^{3}}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)} - \cos x \leadsto \frac{\color{blue}{{\left(\cos \varepsilon \cdot \cos x\right)}^3 - {\left(\sin x \cdot \sin \varepsilon\right)}^3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)} - \cos x\]
      1.1
  1. Removed slow pow expressions

Original test:


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