\[\cos \left(x + \varepsilon\right) - \cos x\]
Test:
NMSE problem 3.3.5
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 30.8 s
Input Error: 39.8
Output Error: 0.7
Log: ⚲
Profile: 🕒
\(\begin{cases} \left(\cos x \cdot \cos \varepsilon - (e^{\log_* (1 + \sin x \cdot \sin \varepsilon)} - 1)^*\right) - \cos x & \text{when } \varepsilon \le -7.175399107131671 \cdot 10^{-12} \\ (\left(-\sin x\right) * \left(\sin \varepsilon\right) + \left({\varepsilon}^2 \cdot \left(-\frac{1}{2}\right)\right))_* & \text{when } \varepsilon \le 17.12856537901564 \\ \left(\cos x \cdot \cos \varepsilon - (e^{\log_* (1 + \sin x \cdot \sin \varepsilon)} - 1)^*\right) - \cos x & \text{otherwise} \end{cases}\)

    if eps < -7.175399107131671e-12 or 17.12856537901564 < eps

    1. Started with
      \[\cos \left(x + \varepsilon\right) - \cos x\]
      31.1
    2. Using strategy rm
      31.1
    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 expm1-log1p-u to get
      \[\left(\cos x \cdot \cos \varepsilon - \color{red}{\sin x \cdot \sin \varepsilon}\right) - \cos x \leadsto \left(\cos x \cdot \cos \varepsilon - \color{blue}{(e^{\log_* (1 + \sin x \cdot \sin \varepsilon)} - 1)^*}\right) - \cos x\]
      1.2

    if -7.175399107131671e-12 < eps < 17.12856537901564

    1. Started with
      \[\cos \left(x + \varepsilon\right) - \cos x\]
      48.8
    2. Using strategy rm
      48.8
    3. Applied cos-sum to get
      \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
      48.4
    4. Using strategy rm
      48.4
    5. Applied expm1-log1p-u to get
      \[\left(\cos x \cdot \cos \varepsilon - \color{red}{\sin x \cdot \sin \varepsilon}\right) - \cos x \leadsto \left(\cos x \cdot \cos \varepsilon - \color{blue}{(e^{\log_* (1 + \sin x \cdot \sin \varepsilon)} - 1)^*}\right) - \cos x\]
      48.4
    6. Applied taylor to get
      \[\left(\cos x \cdot \cos \varepsilon - (e^{\log_* (1 + \sin x \cdot \sin \varepsilon)} - 1)^*\right) - \cos x \leadsto -\left(\frac{1}{2} \cdot {\varepsilon}^2 + (e^{\log_* (1 + \sin x \cdot \sin \varepsilon)} - 1)^*\right)\]
      0.1
    7. Taylor expanded around 0 to get
      \[\color{red}{-\left(\frac{1}{2} \cdot {\varepsilon}^2 + (e^{\log_* (1 + \sin x \cdot \sin \varepsilon)} - 1)^*\right)} \leadsto \color{blue}{-\left(\frac{1}{2} \cdot {\varepsilon}^2 + (e^{\log_* (1 + \sin x \cdot \sin \varepsilon)} - 1)^*\right)}\]
      0.1
    8. Applied simplify to get
      \[\color{red}{-\left(\frac{1}{2} \cdot {\varepsilon}^2 + (e^{\log_* (1 + \sin x \cdot \sin \varepsilon)} - 1)^*\right)} \leadsto \color{blue}{(\left(-\sin x\right) * \left(\sin \varepsilon\right) + \left(\left(-\varepsilon\right) \cdot \left(\varepsilon \cdot \frac{1}{2}\right)\right))_*}\]
      0.1
    9. Applied simplify to get
      \[(\left(-\sin x\right) * \left(\sin \varepsilon\right) + \color{red}{\left(\left(-\varepsilon\right) \cdot \left(\varepsilon \cdot \frac{1}{2}\right)\right)})_* \leadsto (\left(-\sin x\right) * \left(\sin \varepsilon\right) + \color{blue}{\left({\varepsilon}^2 \cdot \left(-\frac{1}{2}\right)\right)})_*\]
      0.1
  1. Removed slow pow expressions

Original test:


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