\[\sin \left(x + \varepsilon\right) - \sin x\]
Test:
NMSE example 3.3
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 19.8 s
Input Error: 37.4
Output Error: 0.4
Log: ⚲
Profile: 🕒
\(\cos x \cdot \sin \varepsilon + \left(\cos \varepsilon \cdot \sin x - \sin x\right)\)
  1. Started with
    \[\sin \left(x + \varepsilon\right) - \sin x\]
    37.4
  2. Using strategy rm
    37.4
  3. Applied sin-sum to get
    \[\color{red}{\sin \left(x + \varepsilon\right)} - \sin x \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
    22.2
  4. Using strategy rm
    22.2
  5. Applied add-cube-cbrt to get
    \[\left(\color{red}{\sin x \cdot \cos \varepsilon} + \cos x \cdot \sin \varepsilon\right) - \sin x \leadsto \left(\color{blue}{{\left(\sqrt[3]{\sin x \cdot \cos \varepsilon}\right)}^3} + \cos x \cdot \sin \varepsilon\right) - \sin x\]
    22.7
  6. Applied taylor to get
    \[\left({\left(\sqrt[3]{\sin x \cdot \cos \varepsilon}\right)}^3 + \cos x \cdot \sin \varepsilon\right) - \sin x \leadsto \left({\left(\sqrt[3]{\sin x \cdot \cos \varepsilon}\right)}^3 + \cos x \cdot \sin \varepsilon\right) - \sin x\]
    22.7
  7. Taylor expanded around 0 to get
    \[\left({\color{red}{\left(\sqrt[3]{\sin x \cdot \cos \varepsilon}\right)}}^3 + \cos x \cdot \sin \varepsilon\right) - \sin x \leadsto \left({\color{blue}{\left(\sqrt[3]{\sin x \cdot \cos \varepsilon}\right)}}^3 + \cos x \cdot \sin \varepsilon\right) - \sin x\]
    22.7
  8. Applied simplify to get
    \[\color{red}{\left({\left(\sqrt[3]{\sin x \cdot \cos \varepsilon}\right)}^3 + \cos x \cdot \sin \varepsilon\right) - \sin x} \leadsto \color{blue}{\cos x \cdot \sin \varepsilon + \left(\cos \varepsilon \cdot \sin x - \sin x\right)}\]
    0.4

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)))))