\[\sin \left(x + \varepsilon\right) - \sin x\]
Test:
NMSE example 3.3
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 8.0 s
Input Error: 16.8
Output Error: 6.3
Log: ⚲
Profile: 🕒
\({\left(\sin x \cdot \cos \varepsilon\right)}^{1} + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\)
  1. Started with
    \[\sin \left(x + \varepsilon\right) - \sin x\]
    16.8
  2. Using strategy rm
    16.8
  3. Applied sin-sum to get
    \[\color{red}{\sin \left(x + \varepsilon\right)} - \sin x \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
    6.3
  4. Applied associate--l+ to get
    \[\color{red}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x} \leadsto \color{blue}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)}\]
    6.3
  5. Using strategy rm
    6.3
  6. Applied pow1 to get
    \[\sin x \cdot \color{red}{\cos \varepsilon} + \left(\cos x \cdot \sin \varepsilon - \sin x\right) \leadsto \sin x \cdot \color{blue}{{\left(\cos \varepsilon\right)}^{1}} + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\]
    6.3
  7. Applied pow1 to get
    \[\color{red}{\sin x} \cdot {\left(\cos \varepsilon\right)}^{1} + \left(\cos x \cdot \sin \varepsilon - \sin x\right) \leadsto \color{blue}{{\left(\sin x\right)}^{1}} \cdot {\left(\cos \varepsilon\right)}^{1} + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\]
    6.3
  8. Applied pow-prod-down to get
    \[\color{red}{{\left(\sin x\right)}^{1} \cdot {\left(\cos \varepsilon\right)}^{1}} + \left(\cos x \cdot \sin \varepsilon - \sin x\right) \leadsto \color{blue}{{\left(\sin x \cdot \cos \varepsilon\right)}^{1}} + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\]
    6.3

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)))))