\[\sin \left(x + \varepsilon\right) - \sin x\]
Test:
NMSE example 3.3
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 23.8 s
Input Error: 16.6
Output Error: 1.5
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{\frac{{\left({\left(\sin x \cdot \cos \varepsilon\right)}^2\right)}^2 - {\left({\left(\cos x \cdot \sin \varepsilon - \sin x\right)}^2\right)}^2}{{\left(\sin x \cdot \cos \varepsilon\right)}^2 + {\left(\cos x \cdot \sin \varepsilon - \sin x\right)}^2}}{(\left(\cos \varepsilon\right) * \left(\sin x\right) + \left(\sin x\right))_* - \sin \varepsilon \cdot \cos x} & \text{when } \varepsilon \le -3.6129147f-16 \\ \frac{\sin x \cdot \left(\cos x \cdot \left(2 \cdot \varepsilon\right)\right) - (\left(\left(\varepsilon \cdot \frac{1}{3}\right) \cdot {\varepsilon}^2\right) * \left(\cos x \cdot \sin x\right) + \left({\varepsilon}^2\right))_*}{(\left(\cos \varepsilon\right) * \left(\sin x\right) + \left(\sin x\right))_* - \cos x \cdot \sin \varepsilon} & \text{when } \varepsilon \le 2.7825011f-09 \\ \frac{\frac{{\left({\left(\sin x \cdot \cos \varepsilon\right)}^2\right)}^2 - {\left({\left(\cos x \cdot \sin \varepsilon - \sin x\right)}^2\right)}^2}{{\left(\sin x \cdot \cos \varepsilon\right)}^2 + {\left(\cos x \cdot \sin \varepsilon - \sin x\right)}^2}}{(\left(\cos \varepsilon\right) * \left(\sin x\right) + \left(\sin x\right))_* - \sin \varepsilon \cdot \cos x} & \text{otherwise} \end{cases}\)

    if eps < -3.6129147f-16 or 2.7825011f-09 < eps

    1. Started with
      \[\sin \left(x + \varepsilon\right) - \sin x\]
      14.2
    2. Using strategy rm
      14.2
    3. Applied sin-sum to get
      \[\color{red}{\sin \left(x + \varepsilon\right)} - \sin x \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
      0.9
    4. Applied associate--l+ to get
      \[\color{red}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x} \leadsto \color{blue}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)}\]
      0.9
    5. Using strategy rm
      0.9
    6. Applied flip-+ to get
      \[\color{red}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)} \leadsto \color{blue}{\frac{{\left(\sin x \cdot \cos \varepsilon\right)}^2 - {\left(\cos x \cdot \sin \varepsilon - \sin x\right)}^2}{\sin x \cdot \cos \varepsilon - \left(\cos x \cdot \sin \varepsilon - \sin x\right)}}\]
      0.9
    7. Applied simplify to get
      \[\frac{{\left(\sin x \cdot \cos \varepsilon\right)}^2 - {\left(\cos x \cdot \sin \varepsilon - \sin x\right)}^2}{\color{red}{\sin x \cdot \cos \varepsilon - \left(\cos x \cdot \sin \varepsilon - \sin x\right)}} \leadsto \frac{{\left(\sin x \cdot \cos \varepsilon\right)}^2 - {\left(\cos x \cdot \sin \varepsilon - \sin x\right)}^2}{\color{blue}{(\left(\cos \varepsilon\right) * \left(\sin x\right) + \left(\sin x\right))_* - \sin \varepsilon \cdot \cos x}}\]
      0.9
    8. Using strategy rm
      0.9
    9. Applied flip-- to get
      \[\frac{\color{red}{{\left(\sin x \cdot \cos \varepsilon\right)}^2 - {\left(\cos x \cdot \sin \varepsilon - \sin x\right)}^2}}{(\left(\cos \varepsilon\right) * \left(\sin x\right) + \left(\sin x\right))_* - \sin \varepsilon \cdot \cos x} \leadsto \frac{\color{blue}{\frac{{\left({\left(\sin x \cdot \cos \varepsilon\right)}^2\right)}^2 - {\left({\left(\cos x \cdot \sin \varepsilon - \sin x\right)}^2\right)}^2}{{\left(\sin x \cdot \cos \varepsilon\right)}^2 + {\left(\cos x \cdot \sin \varepsilon - \sin x\right)}^2}}}{(\left(\cos \varepsilon\right) * \left(\sin x\right) + \left(\sin x\right))_* - \sin \varepsilon \cdot \cos x}\]
      0.9

    if -3.6129147f-16 < eps < 2.7825011f-09

    1. Started with
      \[\sin \left(x + \varepsilon\right) - \sin x\]
      21.1
    2. Using strategy rm
      21.1
    3. Applied sin-sum to get
      \[\color{red}{\sin \left(x + \varepsilon\right)} - \sin x \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
      16.6
    4. Applied associate--l+ to get
      \[\color{red}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x} \leadsto \color{blue}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)}\]
      16.6
    5. Using strategy rm
      16.6
    6. Applied flip-+ to get
      \[\color{red}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)} \leadsto \color{blue}{\frac{{\left(\sin x \cdot \cos \varepsilon\right)}^2 - {\left(\cos x \cdot \sin \varepsilon - \sin x\right)}^2}{\sin x \cdot \cos \varepsilon - \left(\cos x \cdot \sin \varepsilon - \sin x\right)}}\]
      16.6
    7. Applied simplify to get
      \[\frac{{\left(\sin x \cdot \cos \varepsilon\right)}^2 - {\left(\cos x \cdot \sin \varepsilon - \sin x\right)}^2}{\color{red}{\sin x \cdot \cos \varepsilon - \left(\cos x \cdot \sin \varepsilon - \sin x\right)}} \leadsto \frac{{\left(\sin x \cdot \cos \varepsilon\right)}^2 - {\left(\cos x \cdot \sin \varepsilon - \sin x\right)}^2}{\color{blue}{(\left(\cos \varepsilon\right) * \left(\sin x\right) + \left(\sin x\right))_* - \sin \varepsilon \cdot \cos x}}\]
      16.6
    8. Using strategy rm
      16.6
    9. Applied add-cube-cbrt to get
      \[\frac{\color{red}{{\left(\sin x \cdot \cos \varepsilon\right)}^2 - {\left(\cos x \cdot \sin \varepsilon - \sin x\right)}^2}}{(\left(\cos \varepsilon\right) * \left(\sin x\right) + \left(\sin x\right))_* - \sin \varepsilon \cdot \cos x} \leadsto \frac{\color{blue}{{\left(\sqrt[3]{{\left(\sin x \cdot \cos \varepsilon\right)}^2 - {\left(\cos x \cdot \sin \varepsilon - \sin x\right)}^2}\right)}^3}}{(\left(\cos \varepsilon\right) * \left(\sin x\right) + \left(\sin x\right))_* - \sin \varepsilon \cdot \cos x}\]
      21.1
    10. Applied taylor to get
      \[\frac{{\left(\sqrt[3]{{\left(\sin x \cdot \cos \varepsilon\right)}^2 - {\left(\cos x \cdot \sin \varepsilon - \sin x\right)}^2}\right)}^3}{(\left(\cos \varepsilon\right) * \left(\sin x\right) + \left(\sin x\right))_* - \sin \varepsilon \cdot \cos x} \leadsto \frac{2 \cdot \left(\varepsilon \cdot \left(\sin x \cdot \cos x\right)\right) - \left(\frac{1}{3} \cdot \left({\varepsilon}^{3} \cdot \left(\sin x \cdot \cos x\right)\right) + \left({\varepsilon}^2 \cdot {\left(\sin x\right)}^2 + {\varepsilon}^2 \cdot {\left(\cos x\right)}^2\right)\right)}{(\left(\cos \varepsilon\right) * \left(\sin x\right) + \left(\sin x\right))_* - \sin \varepsilon \cdot \cos x}\]
      2.7
    11. Taylor expanded around 0 to get
      \[\frac{\color{red}{2 \cdot \left(\varepsilon \cdot \left(\sin x \cdot \cos x\right)\right) - \left(\frac{1}{3} \cdot \left({\varepsilon}^{3} \cdot \left(\sin x \cdot \cos x\right)\right) + \left({\varepsilon}^2 \cdot {\left(\sin x\right)}^2 + {\varepsilon}^2 \cdot {\left(\cos x\right)}^2\right)\right)}}{(\left(\cos \varepsilon\right) * \left(\sin x\right) + \left(\sin x\right))_* - \sin \varepsilon \cdot \cos x} \leadsto \frac{\color{blue}{2 \cdot \left(\varepsilon \cdot \left(\sin x \cdot \cos x\right)\right) - \left(\frac{1}{3} \cdot \left({\varepsilon}^{3} \cdot \left(\sin x \cdot \cos x\right)\right) + \left({\varepsilon}^2 \cdot {\left(\sin x\right)}^2 + {\varepsilon}^2 \cdot {\left(\cos x\right)}^2\right)\right)}}{(\left(\cos \varepsilon\right) * \left(\sin x\right) + \left(\sin x\right))_* - \sin \varepsilon \cdot \cos x}\]
      2.7
    12. Applied simplify to get
      \[\frac{2 \cdot \left(\varepsilon \cdot \left(\sin x \cdot \cos x\right)\right) - \left(\frac{1}{3} \cdot \left({\varepsilon}^{3} \cdot \left(\sin x \cdot \cos x\right)\right) + \left({\varepsilon}^2 \cdot {\left(\sin x\right)}^2 + {\varepsilon}^2 \cdot {\left(\cos x\right)}^2\right)\right)}{(\left(\cos \varepsilon\right) * \left(\sin x\right) + \left(\sin x\right))_* - \sin \varepsilon \cdot \cos x} \leadsto \frac{\left(\left(\varepsilon \cdot 2\right) \cdot \cos x\right) \cdot \sin x - (\left(\left(\frac{1}{3} \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) * \left(\cos x \cdot \sin x\right) + \left(\left(\varepsilon \cdot \varepsilon\right) \cdot 1\right))_*}{(\left(\cos \varepsilon\right) * \left(\sin x\right) + \left(\sin x\right))_* - \cos x \cdot \sin \varepsilon}\]
      2.7
    13. Applied final simplification
    14. Applied simplify to get
      \[\color{red}{\frac{\left(\left(\varepsilon \cdot 2\right) \cdot \cos x\right) \cdot \sin x - (\left(\left(\frac{1}{3} \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) * \left(\cos x \cdot \sin x\right) + \left(\left(\varepsilon \cdot \varepsilon\right) \cdot 1\right))_*}{(\left(\cos \varepsilon\right) * \left(\sin x\right) + \left(\sin x\right))_* - \cos x \cdot \sin \varepsilon}} \leadsto \color{blue}{\frac{\sin x \cdot \left(\cos x \cdot \left(2 \cdot \varepsilon\right)\right) - (\left(\left(\varepsilon \cdot \frac{1}{3}\right) \cdot {\varepsilon}^2\right) * \left(\cos x \cdot \sin x\right) + \left({\varepsilon}^2\right))_*}{(\left(\cos \varepsilon\right) * \left(\sin x\right) + \left(\sin x\right))_* - \cos x \cdot \sin \varepsilon}}\]
      2.7
  1. 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)))))