\[\sin \left(x + \varepsilon\right) - \sin x\]
Test:
NMSE example 3.3
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 23.9 s
Input Error: 36.5
Output Error: 0.5
Log: ⚲
Profile: 🕒
\(\log_* (1 + (e^{(\left(\sin \varepsilon\right) * \left(\cos x\right) + \left(\cos \varepsilon \cdot \sin x - \sin x\right))_*} - 1)^*)\)
  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 log1p-expm1-u to get
    \[\color{red}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x} \leadsto \color{blue}{\log_* (1 + (e^{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x} - 1)^*)}\]
    21.8
  6. Applied simplify to get
    \[\log_* (1 + \color{red}{(e^{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x} - 1)^*}) \leadsto \log_* (1 + \color{blue}{(e^{(\left(\sin \varepsilon\right) * \left(\cos x\right) + \left(\cos \varepsilon \cdot \sin x - \sin x\right))_*} - 1)^*})\]
    0.5
  7. Removed slow pow expressions

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)))))