\[\sin \left(x + \varepsilon\right) - \sin x\]
Test:
NMSE example 3.3
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 28.8 s
Input Error: 36.8
Output Error: 0.5
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{{\left(\sin \varepsilon \cdot \cos x + \cos \varepsilon \cdot \sin x\right)}^3 - {\left(\sin x\right)}^3}{{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)}^2 + \left({\left(\sin x\right)}^2 + \left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) \cdot \sin x\right)} & \text{when } \varepsilon \le -9.280756734832815 \cdot 10^{-07} \\ \left(\varepsilon - \frac{1}{6} \cdot {\varepsilon}^3\right) \cdot \cos x - \left(\sin x \cdot \frac{1}{2}\right) \cdot {\varepsilon}^2 & \text{when } \varepsilon \le 1.0802375057316329 \cdot 10^{-14} \\ \frac{{\left(\sin \varepsilon \cdot \cos x + \cos \varepsilon \cdot \sin x\right)}^3 - {\left(\sin x\right)}^3}{{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)}^2 + \left({\left(\sin x\right)}^2 + \left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) \cdot \sin x\right)} & \text{otherwise} \end{cases}\)

    if eps < -9.280756734832815e-07 or 1.0802375057316329e-14 < eps

    1. Started with
      \[\sin \left(x + \varepsilon\right) - \sin x\]
      29.4
    2. Using strategy rm
      29.4
    3. Applied sin-sum to get
      \[\color{red}{\sin \left(x + \varepsilon\right)} - \sin x \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
      0.7
    4. Using strategy rm
      0.7
    5. Applied flip3-- to get
      \[\color{red}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x} \leadsto \color{blue}{\frac{{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)}^{3} - {\left(\sin x\right)}^{3}}{{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)}^2 + \left({\left(\sin x\right)}^2 + \left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) \cdot \sin x\right)}}\]
      0.9
    6. Applied simplify to get
      \[\frac{\color{red}{{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)}^{3} - {\left(\sin x\right)}^{3}}}{{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)}^2 + \left({\left(\sin x\right)}^2 + \left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) \cdot \sin x\right)} \leadsto \frac{\color{blue}{{\left(\sin \varepsilon \cdot \cos x + \cos \varepsilon \cdot \sin x\right)}^3 - {\left(\sin x\right)}^3}}{{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)}^2 + \left({\left(\sin x\right)}^2 + \left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) \cdot \sin x\right)}\]
      0.8

    if -9.280756734832815e-07 < eps < 1.0802375057316329e-14

    1. Started with
      \[\sin \left(x + \varepsilon\right) - \sin x\]
      44.9
    2. Using strategy rm
      44.9
    3. Applied sin-sum to get
      \[\color{red}{\sin \left(x + \varepsilon\right)} - \sin x \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
      44.6
    4. Using strategy rm
      44.6
    5. Applied add-cube-cbrt to get
      \[\color{red}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x} \leadsto \color{blue}{{\left(\sqrt[3]{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x}\right)}^3}\]
      45.0
    6. Applied taylor to get
      \[{\left(\sqrt[3]{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x}\right)}^3 \leadsto \varepsilon \cdot \cos x - \left(\frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + \frac{1}{2} \cdot \left({\varepsilon}^2 \cdot \sin x\right)\right)\]
      0.1
    7. Taylor expanded around 0 to get
      \[\color{red}{\varepsilon \cdot \cos x - \left(\frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + \frac{1}{2} \cdot \left({\varepsilon}^2 \cdot \sin x\right)\right)} \leadsto \color{blue}{\varepsilon \cdot \cos x - \left(\frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + \frac{1}{2} \cdot \left({\varepsilon}^2 \cdot \sin x\right)\right)}\]
      0.1
    8. Applied simplify to get
      \[\varepsilon \cdot \cos x - \left(\frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + \frac{1}{2} \cdot \left({\varepsilon}^2 \cdot \sin x\right)\right) \leadsto \left(\varepsilon - \frac{1}{6} \cdot {\varepsilon}^3\right) \cdot \cos x - \left(\sin x \cdot \frac{1}{2}\right) \cdot {\varepsilon}^2\]
      0.1
    9. Applied final simplification
  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)))))