\[\cos \left(x + \varepsilon\right) - \cos x\]
Test:
NMSE problem 3.3.5
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 15.7 s
Input Error: 18.2
Output Error: 0.8
Log: ⚲
Profile: 🕒
\(\begin{cases} \left(\cos x \cdot \cos \varepsilon - \log_* (1 + (e^{\sin x \cdot \sin \varepsilon} - 1)^*)\right) - \cos x & \text{when } \varepsilon \le -0.010431706f0 \\ -(\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \frac{1}{2}\right))_* & \text{when } \varepsilon \le 3.9492395f-05 \\ \left(\cos x \cdot \cos \varepsilon - \log_* (1 + (e^{\sin x \cdot \sin \varepsilon} - 1)^*)\right) - \cos x & \text{otherwise} \end{cases}\)

    if eps < -0.010431706f0 or 3.9492395f-05 < eps

    1. Started with
      \[\cos \left(x + \varepsilon\right) - \cos x\]
      14.4
    2. Using strategy rm
      14.4
    3. Applied cos-sum to get
      \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
      1.2
    4. Using strategy rm
      1.2
    5. Applied log1p-expm1-u to get
      \[\left(\cos x \cdot \cos \varepsilon - \color{red}{\sin x \cdot \sin \varepsilon}\right) - \cos x \leadsto \left(\cos x \cdot \cos \varepsilon - \color{blue}{\log_* (1 + (e^{\sin x \cdot \sin \varepsilon} - 1)^*)}\right) - \cos x\]
      1.3

    if -0.010431706f0 < eps < 3.9492395f-05

    1. Started with
      \[\cos \left(x + \varepsilon\right) - \cos x\]
      22.8
    2. Using strategy rm
      22.8
    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
      \[\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \color{red}{\sin \varepsilon}\right) - \cos x \leadsto \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \color{blue}{{\left(\sqrt[3]{\sin \varepsilon}\right)}^3}\right) - \cos x\]
      18.7
    6. Applied add-cube-cbrt to get
      \[\left(\cos x \cdot \cos \varepsilon - \color{red}{\sin x} \cdot {\left(\sqrt[3]{\sin \varepsilon}\right)}^3\right) - \cos x \leadsto \left(\cos x \cdot \cos \varepsilon - \color{blue}{{\left(\sqrt[3]{\sin x}\right)}^3} \cdot {\left(\sqrt[3]{\sin \varepsilon}\right)}^3\right) - \cos x\]
      18.7
    7. Applied cube-unprod to get
      \[\left(\cos x \cdot \cos \varepsilon - \color{red}{{\left(\sqrt[3]{\sin x}\right)}^3 \cdot {\left(\sqrt[3]{\sin \varepsilon}\right)}^3}\right) - \cos x \leadsto \left(\cos x \cdot \cos \varepsilon - \color{blue}{{\left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin \varepsilon}\right)}^3}\right) - \cos x\]
      18.7
    8. Applied taylor to get
      \[\left(\cos x \cdot \cos \varepsilon - {\left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin \varepsilon}\right)}^3\right) - \cos x \leadsto -\left(\frac{1}{2} \cdot {\varepsilon}^2 + {\left(\sqrt[3]{\sin \varepsilon} \cdot \sqrt[3]{\sin x}\right)}^3\right)\]
      0.6
    9. Taylor expanded around 0 to get
      \[\color{red}{-\left(\frac{1}{2} \cdot {\varepsilon}^2 + {\left(\sqrt[3]{\sin \varepsilon} \cdot \sqrt[3]{\sin x}\right)}^3\right)} \leadsto \color{blue}{-\left(\frac{1}{2} \cdot {\varepsilon}^2 + {\left(\sqrt[3]{\sin \varepsilon} \cdot \sqrt[3]{\sin x}\right)}^3\right)}\]
      0.6
    10. Applied simplify to get
      \[-\left(\frac{1}{2} \cdot {\varepsilon}^2 + {\left(\sqrt[3]{\sin \varepsilon} \cdot \sqrt[3]{\sin x}\right)}^3\right) \leadsto -(\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \frac{1}{2}\right))_*\]
      0.2
    11. 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)))