\[\sin \left(x + \varepsilon\right) - \sin x\]
Test:
NMSE example 3.3
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 4.0 s
Input Error: 17.5
Output Error: 0.1
Log: ⚲
Profile: 🕒
\(\cos x \cdot \sin \varepsilon\)
  1. Started with
    \[\sin \left(x + \varepsilon\right) - \sin x\]
    17.5
  2. Using strategy rm
    17.5
  3. Applied sin-sum to get
    \[\color{red}{\sin \left(x + \varepsilon\right)} - \sin x \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
    13.2
  4. Applied associate--l+ to get
    \[\color{red}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x} \leadsto \color{blue}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)}\]
    13.2
  5. Using strategy rm
    13.2
  6. Applied log1p-expm1-u to get
    \[\sin x \cdot \cos \varepsilon + \left(\color{red}{\cos x \cdot \sin \varepsilon} - \sin x\right) \leadsto \sin x \cdot \cos \varepsilon + \left(\color{blue}{\log_* (1 + (e^{\cos x \cdot \sin \varepsilon} - 1)^*)} - \sin x\right)\]
    13.3
  7. Applied taylor to get
    \[\sin x \cdot \cos \varepsilon + \left(\log_* (1 + (e^{\cos x \cdot \sin \varepsilon} - 1)^*) - \sin x\right) \leadsto \log_* (1 + (e^{\sin \varepsilon \cdot \cos x} - 1)^*)\]
    0.2
  8. Taylor expanded around 0 to get
    \[\color{red}{\log_* (1 + (e^{\sin \varepsilon \cdot \cos x} - 1)^*)} \leadsto \color{blue}{\log_* (1 + (e^{\sin \varepsilon \cdot \cos x} - 1)^*)}\]
    0.2
  9. Applied simplify to get
    \[\color{red}{\log_* (1 + (e^{\sin \varepsilon \cdot \cos x} - 1)^*)} \leadsto \color{blue}{\cos x \cdot \sin \varepsilon}\]
    0.1

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)))))