\[\sin \left(x + \varepsilon\right) - \sin x\]
Test:
NMSE example 3.3
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 22.4 s
Input Error: 36.5
Output Error: 0.4
Log: ⚲
Profile: 🕒
\(\cos x \cdot \sin \varepsilon + \left(\cos \varepsilon \cdot \sin x - \sin x\right)\)
  1. Started with
    \[\sin \left(x + \varepsilon\right) - \sin x\]
    36.5
  2. Using strategy rm
    36.5
  3. Applied sin-sum to get
    \[\color{red}{\sin \left(x + \varepsilon\right)} - \sin x \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
    21.7
  4. Using strategy rm
    21.7
  5. Applied add-cube-cbrt to get
    \[\left(\sin x \cdot \cos \varepsilon + \color{red}{\cos x \cdot \sin \varepsilon}\right) - \sin x \leadsto \left(\sin x \cdot \cos \varepsilon + \color{blue}{{\left(\sqrt[3]{\cos x \cdot \sin \varepsilon}\right)}^3}\right) - \sin x\]
    22.4
  6. Applied taylor to get
    \[\left(\sin x \cdot \cos \varepsilon + {\left(\sqrt[3]{\cos x \cdot \sin \varepsilon}\right)}^3\right) - \sin x \leadsto \left(\sin x \cdot \cos \varepsilon + {\left(\sqrt[3]{\sin \varepsilon \cdot \cos x}\right)}^3\right) - \sin x\]
    22.4
  7. Taylor expanded around 0 to get
    \[\left(\sin x \cdot \cos \varepsilon + {\color{red}{\left(\sqrt[3]{\sin \varepsilon \cdot \cos x}\right)}}^3\right) - \sin x \leadsto \left(\sin x \cdot \cos \varepsilon + {\color{blue}{\left(\sqrt[3]{\sin \varepsilon \cdot \cos x}\right)}}^3\right) - \sin x\]
    22.4
  8. Applied simplify to get
    \[\left(\sin x \cdot \cos \varepsilon + {\left(\sqrt[3]{\sin \varepsilon \cdot \cos x}\right)}^3\right) - \sin x \leadsto \cos x \cdot \sin \varepsilon + \left(\cos \varepsilon \cdot \sin x - \sin x\right)\]
    0.4
  9. 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)))))