\[\sin \left(x + \varepsilon\right) - \sin x\]
Test:
NMSE example 3.3
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 24.2 s
Input Error: 37.4
Output Error: 0.3
Log: ⚲
Profile: 🕒
\((\left(\sin \varepsilon\right) * \left(\cos x\right) + \left(\left(\cos \varepsilon - 1\right) \cdot \sin x\right))_*\)
  1. Started with
    \[\sin \left(x + \varepsilon\right) - \sin x\]
    37.4
  2. Using strategy rm
    37.4
  3. Applied sin-sum to get
    \[\color{red}{\sin \left(x + \varepsilon\right)} - \sin x \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
    22.2
  4. Using strategy rm
    22.2
  5. Applied add-cube-cbrt to get
    \[\color{red}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x} \leadsto \color{blue}{{\left(\sqrt[3]{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x}\right)}^3}\]
    22.9
  6. Applied simplify to get
    \[{\color{red}{\left(\sqrt[3]{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x}\right)}}^3 \leadsto {\color{blue}{\left(\sqrt[3]{(\left(\sin \varepsilon\right) * \left(\cos x\right) + \left(\cos \varepsilon \cdot \sin x - \sin x\right))_*}\right)}}^3\]
    1.6
  7. Using strategy rm
    1.6
  8. Applied *-un-lft-identity to get
    \[{\left(\sqrt[3]{(\left(\sin \varepsilon\right) * \left(\cos x\right) + \left(\cos \varepsilon \cdot \sin x - \color{red}{\sin x}\right))_*}\right)}^3 \leadsto {\left(\sqrt[3]{(\left(\sin \varepsilon\right) * \left(\cos x\right) + \left(\cos \varepsilon \cdot \sin x - \color{blue}{1 \cdot \sin x}\right))_*}\right)}^3\]
    1.6
  9. Applied distribute-rgt-out-- to get
    \[{\left(\sqrt[3]{(\left(\sin \varepsilon\right) * \left(\cos x\right) + \color{red}{\left(\cos \varepsilon \cdot \sin x - 1 \cdot \sin x\right)})_*}\right)}^3 \leadsto {\left(\sqrt[3]{(\left(\sin \varepsilon\right) * \left(\cos x\right) + \color{blue}{\left(\sin x \cdot \left(\cos \varepsilon - 1\right)\right)})_*}\right)}^3\]
    1.6
  10. Applied taylor to get
    \[{\left(\sqrt[3]{(\left(\sin \varepsilon\right) * \left(\cos x\right) + \left(\sin x \cdot \left(\cos \varepsilon - 1\right)\right))_*}\right)}^3 \leadsto (\left(\sin \varepsilon\right) * \left(\cos x\right) + \left(\sin x \cdot \left(\cos \varepsilon - 1\right)\right))_*\]
    0.3
  11. Taylor expanded around 0 to get
    \[\color{red}{(\left(\sin \varepsilon\right) * \left(\cos x\right) + \left(\sin x \cdot \left(\cos \varepsilon - 1\right)\right))_*} \leadsto \color{blue}{(\left(\sin \varepsilon\right) * \left(\cos x\right) + \left(\sin x \cdot \left(\cos \varepsilon - 1\right)\right))_*}\]
    0.3
  12. Applied simplify to get
    \[(\left(\sin \varepsilon\right) * \left(\cos x\right) + \left(\sin x \cdot \left(\cos \varepsilon - 1\right)\right))_* \leadsto (\left(\sin \varepsilon\right) * \left(\cos x\right) + \left(\left(\cos \varepsilon - 1\right) \cdot \sin x\right))_*\]
    0.3
  13. Applied final simplification
  14. 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)))))