\[\cos \left(x + \varepsilon\right) - \cos x\]
Test:
NMSE problem 3.3.5
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 16.6 s
Input Error: 18.4
Output Error: 0.7
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{\log_* (1 + (e^{{\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right)}^3} - 1)^*) - {\left(\cos x\right)}^3}{(\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) * \left((\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin \varepsilon \cdot \sin x\right) + \left(\cos x \cdot \cos x\right))_*} & \text{when } \varepsilon \le -0.0018968344f0 \\ {\varepsilon}^3 \cdot \left(\frac{1}{6} \cdot \sin x\right) - \varepsilon \cdot (\frac{1}{2} * \varepsilon + \left(\sin x\right))_* & \text{when } \varepsilon \le 0.0018547908f0 \\ \frac{\log_* (1 + (e^{{\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right)}^3} - 1)^*) - {\left(\cos x\right)}^3}{(\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) * \left((\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin \varepsilon \cdot \sin x\right) + \left(\cos x \cdot \cos x\right))_*} & \text{otherwise} \end{cases}\)

    if eps < -0.0018968344f0

    1. Started with
      \[\cos \left(x + \varepsilon\right) - \cos x\]
      14.4
    2. Using strategy rm
      14.4
    3. Applied cos-sum to get
      \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
      1.0
    4. Using strategy rm
      1.0
    5. Applied flip3-- to get
      \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x} \leadsto \color{blue}{\frac{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^{3} - {\left(\cos x\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^2 + \left({\left(\cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) \cdot \cos x\right)}}\]
      1.1
    6. Applied simplify to get
      \[\frac{\color{red}{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^{3} - {\left(\cos x\right)}^{3}}}{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^2 + \left({\left(\cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) \cdot \cos x\right)} \leadsto \frac{\color{blue}{{\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right)}^3 - {\left(\cos x\right)}^3}}{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^2 + \left({\left(\cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) \cdot \cos x\right)}\]
      1.0
    7. Applied simplify to get
      \[\frac{{\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right)}^3 - {\left(\cos x\right)}^3}{\color{red}{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^2 + \left({\left(\cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) \cdot \cos x\right)}} \leadsto \frac{{\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right)}^3 - {\left(\cos x\right)}^3}{\color{blue}{(\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) * \left((\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin \varepsilon \cdot \sin x\right) + \left(\cos x \cdot \cos x\right))_*}}\]
      1.1
    8. Using strategy rm
      1.1
    9. Applied log1p-expm1-u to get
      \[\frac{\color{red}{{\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right)}^3} - {\left(\cos x\right)}^3}{(\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) * \left((\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin \varepsilon \cdot \sin x\right) + \left(\cos x \cdot \cos x\right))_*} \leadsto \frac{\color{blue}{\log_* (1 + (e^{{\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right)}^3} - 1)^*)} - {\left(\cos x\right)}^3}{(\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) * \left((\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin \varepsilon \cdot \sin x\right) + \left(\cos x \cdot \cos x\right))_*}\]
      1.1

    if -0.0018968344f0 < eps < 0.0018547908f0

    1. Started with
      \[\cos \left(x + \varepsilon\right) - \cos x\]
      23.0
    2. Using strategy rm
      23.0
    3. Applied cos-sum to get
      \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
      18.7
    4. Using strategy rm
      18.7
    5. Applied add-cube-cbrt to get
      \[\left(\cos x \cdot \cos \varepsilon - \color{red}{\sin x \cdot \sin \varepsilon}\right) - \cos x \leadsto \left(\cos x \cdot \cos \varepsilon - \color{blue}{{\left(\sqrt[3]{\sin x \cdot \sin \varepsilon}\right)}^3}\right) - \cos x\]
      18.7
    6. Applied taylor to get
      \[\left(\cos x \cdot \cos \varepsilon - {\left(\sqrt[3]{\sin x \cdot \sin \varepsilon}\right)}^3\right) - \cos x \leadsto \frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot \sin x\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot \sin x\right)\]
      0.2
    7. Taylor expanded around 0 to get
      \[\color{red}{\frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot \sin x\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot \sin x\right)} \leadsto \color{blue}{\frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot \sin x\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot \sin x\right)}\]
      0.2
    8. Applied simplify to get
      \[\frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot \sin x\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot \sin x\right) \leadsto \frac{1}{6} \cdot \left({\varepsilon}^3 \cdot \sin x\right) - \varepsilon \cdot \left(\sin x + \frac{1}{2} \cdot \varepsilon\right)\]
      0.1
    9. Applied final simplification
    10. Applied simplify to get
      \[\color{red}{\frac{1}{6} \cdot \left({\varepsilon}^3 \cdot \sin x\right) - \varepsilon \cdot \left(\sin x + \frac{1}{2} \cdot \varepsilon\right)} \leadsto \color{blue}{{\varepsilon}^3 \cdot \left(\frac{1}{6} \cdot \sin x\right) - \varepsilon \cdot (\frac{1}{2} * \varepsilon + \left(\sin x\right))_*}\]
      0.2

    if 0.0018547908f0 < eps

    1. Started with
      \[\cos \left(x + \varepsilon\right) - \cos x\]
      14.2
    2. Using strategy rm
      14.2
    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
      \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x} \leadsto \color{blue}{\frac{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^{3} - {\left(\cos x\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^2 + \left({\left(\cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) \cdot \cos x\right)}}\]
      1.1
    6. Applied simplify to get
      \[\frac{\color{red}{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^{3} - {\left(\cos x\right)}^{3}}}{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^2 + \left({\left(\cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) \cdot \cos x\right)} \leadsto \frac{\color{blue}{{\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right)}^3 - {\left(\cos x\right)}^3}}{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^2 + \left({\left(\cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) \cdot \cos x\right)}\]
      0.9
    7. Applied simplify to get
      \[\frac{{\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right)}^3 - {\left(\cos x\right)}^3}{\color{red}{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^2 + \left({\left(\cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) \cdot \cos x\right)}} \leadsto \frac{{\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right)}^3 - {\left(\cos x\right)}^3}{\color{blue}{(\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) * \left((\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin \varepsilon \cdot \sin x\right) + \left(\cos x \cdot \cos x\right))_*}}\]
      1.1
    8. Using strategy rm
      1.1
    9. Applied log1p-expm1-u to get
      \[\frac{\color{red}{{\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right)}^3} - {\left(\cos x\right)}^3}{(\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) * \left((\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin \varepsilon \cdot \sin x\right) + \left(\cos x \cdot \cos x\right))_*} \leadsto \frac{\color{blue}{\log_* (1 + (e^{{\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right)}^3} - 1)^*)} - {\left(\cos x\right)}^3}{(\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) * \left((\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin \varepsilon \cdot \sin x\right) + \left(\cos x \cdot \cos x\right))_*}\]
      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)))