\[\sin \left(x + \varepsilon\right) - \sin x\]
Test:
NMSE example 3.3
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 7.7 s
Input Error: 16.6
Output Error: 0.5
Log: ⚲
Profile: 🕒
\((\left(\sin \varepsilon\right) * \left(\cos x\right) + \left(\sin x \cdot \cos \varepsilon - \sin x\right))_*\)
  1. Started with
    \[\sin \left(x + \varepsilon\right) - \sin x\]
    16.6
  2. Using strategy rm
    16.6
  3. Applied sin-sum to get
    \[\color{red}{\sin \left(x + \varepsilon\right)} - \sin x \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
    6.1
  4. Applied associate--l+ to get
    \[\color{red}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x} \leadsto \color{blue}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)}\]
    6.1
  5. Using strategy rm
    6.1
  6. Applied add-log-exp to get
    \[\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \color{red}{\sin x}\right) \leadsto \sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \color{blue}{\log \left(e^{\sin x}\right)}\right)\]
    9.6
  7. Applied add-log-exp to get
    \[\sin x \cdot \cos \varepsilon + \left(\color{red}{\cos x \cdot \sin \varepsilon} - \log \left(e^{\sin x}\right)\right) \leadsto \sin x \cdot \cos \varepsilon + \left(\color{blue}{\log \left(e^{\cos x \cdot \sin \varepsilon}\right)} - \log \left(e^{\sin x}\right)\right)\]
    13.0
  8. Applied diff-log to get
    \[\sin x \cdot \cos \varepsilon + \color{red}{\left(\log \left(e^{\cos x \cdot \sin \varepsilon}\right) - \log \left(e^{\sin x}\right)\right)} \leadsto \sin x \cdot \cos \varepsilon + \color{blue}{\log \left(\frac{e^{\cos x \cdot \sin \varepsilon}}{e^{\sin x}}\right)}\]
    13.0
  9. Applied add-log-exp to get
    \[\color{red}{\sin x \cdot \cos \varepsilon} + \log \left(\frac{e^{\cos x \cdot \sin \varepsilon}}{e^{\sin x}}\right) \leadsto \color{blue}{\log \left(e^{\sin x \cdot \cos \varepsilon}\right)} + \log \left(\frac{e^{\cos x \cdot \sin \varepsilon}}{e^{\sin x}}\right)\]
    13.0
  10. Applied sum-log to get
    \[\color{red}{\log \left(e^{\sin x \cdot \cos \varepsilon}\right) + \log \left(\frac{e^{\cos x \cdot \sin \varepsilon}}{e^{\sin x}}\right)} \leadsto \color{blue}{\log \left(e^{\sin x \cdot \cos \varepsilon} \cdot \frac{e^{\cos x \cdot \sin \varepsilon}}{e^{\sin x}}\right)}\]
    12.8
  11. Applied simplify to get
    \[\log \color{red}{\left(e^{\sin x \cdot \cos \varepsilon} \cdot \frac{e^{\cos x \cdot \sin \varepsilon}}{e^{\sin x}}\right)} \leadsto \log \color{blue}{\left(e^{(\left(\sin \varepsilon\right) * \left(\cos x\right) + \left(\cos \varepsilon \cdot \sin x - \sin x\right))_*}\right)}\]
    12.7
  12. Applied taylor to get
    \[\log \left(e^{(\left(\sin \varepsilon\right) * \left(\cos x\right) + \left(\cos \varepsilon \cdot \sin x - \sin x\right))_*}\right) \leadsto (\left(\sin \varepsilon\right) * \left(\cos x\right) + \left(\sin x \cdot \cos \varepsilon - \sin x\right))_*\]
    0.5
  13. Taylor expanded around 0 to get
    \[\color{red}{(\left(\sin \varepsilon\right) * \left(\cos x\right) + \left(\sin x \cdot \cos \varepsilon - \sin x\right))_*} \leadsto \color{blue}{(\left(\sin \varepsilon\right) * \left(\cos x\right) + \left(\sin x \cdot \cos \varepsilon - \sin x\right))_*}\]
    0.5
  14. Applied simplify to get
    \[(\left(\sin \varepsilon\right) * \left(\cos x\right) + \left(\sin x \cdot \cos \varepsilon - \sin x\right))_* \leadsto (\left(\sin \varepsilon\right) * \left(\cos x\right) + \left(\sin x \cdot \cos \varepsilon - \sin x\right))_*\]
    0.5
  15. Applied final simplification

Original test:


(lambda ((x default) (eps default))
  #:name "NMSE example 3.3"
  (- (sin (+ x eps)) (sin x))
  #:target
  (* 2 (* (cos (+ x (/ eps 2))) (sin (/ eps 2)))))