\[\sin \left(x + \varepsilon\right) - \sin x\]
Test:
NMSE example 3.3
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 17.5 s
Input Error: 36.4
Output Error: 21.6
Log: ⚲
Profile: 🕒
\((\left(\cos x\right) * \left(\sin \varepsilon\right) + \left(\cos \varepsilon \cdot \sin x\right))_* + \left(-\sin x\right)\)
  1. Started with
    \[\sin \left(x + \varepsilon\right) - \sin x\]
    36.4
  2. Using strategy rm
    36.4
  3. Applied sin-sum to get
    \[\color{red}{\sin \left(x + \varepsilon\right)} - \sin x \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
    21.6
  4. Applied associate--l+ to get
    \[\color{red}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x} \leadsto \color{blue}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)}\]
    21.6
  5. Using strategy rm
    21.6
  6. Applied sub-neg to get
    \[\sin x \cdot \cos \varepsilon + \color{red}{\left(\cos x \cdot \sin \varepsilon - \sin x\right)} \leadsto \sin x \cdot \cos \varepsilon + \color{blue}{\left(\cos x \cdot \sin \varepsilon + \left(-\sin x\right)\right)}\]
    21.6
  7. Applied associate-+r+ to get
    \[\color{red}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon + \left(-\sin x\right)\right)} \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) + \left(-\sin x\right)}\]
    21.6
  8. Applied simplify to get
    \[\color{red}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} + \left(-\sin x\right) \leadsto \color{blue}{(\left(\cos x\right) * \left(\sin \varepsilon\right) + \left(\cos \varepsilon \cdot \sin x\right))_*} + \left(-\sin x\right)\]
    21.6

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)))))