\[\sin \left(x + \varepsilon\right) - \sin x\]
Test:
NMSE example 3.3
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 1.0 m
Input Error: 36.3
Output Error: 2.1
Log: ⚲
\(\begin{cases} \frac{{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)}^2 - {\left(\sin x\right)}^2}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) + \sin x} & \text{when } \varepsilon \le -6.169079171368681 \cdot 10^{-49} \\ \varepsilon - \left(\left(x + \varepsilon\right) \cdot \left(x \cdot \varepsilon\right)\right) \cdot \frac{1}{2} & \text{when } \varepsilon \le 2.0116360245016707 \cdot 10^{-26} \\ \sin x \cdot \cos \varepsilon + \frac{{\left(\cos x \cdot \sin \varepsilon\right)}^2 - {\left(\sin x\right)}^2}{\cos x \cdot \sin \varepsilon + \sin x} & \text{otherwise} \end{cases}\)

    if eps < -6.169079171368681e-49

    1. Started with
      \[\sin \left(x + \varepsilon\right) - \sin x\]
      30.4
    2. Using strategy rm
      30.4
    3. Applied sin-sum to get
      \[\color{red}{\sin \left(x + \varepsilon\right)} - \sin x \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
      4.4
    4. Using strategy rm
      4.4
    5. Applied flip-- to get
      \[\color{red}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x} \leadsto \color{blue}{\frac{{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)}^2 - {\left(\sin x\right)}^2}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) + \sin x}}\]
      4.6

    if -6.169079171368681e-49 < eps < 2.0116360245016707e-26

    1. Started with
      \[\sin \left(x + \varepsilon\right) - \sin x\]
      44.9
    2. Applied taylor to get
      \[\sin \left(x + \varepsilon\right) - \sin x \leadsto \varepsilon - \left(\frac{1}{2} \cdot \left({x}^2 \cdot \varepsilon\right) + \frac{1}{2} \cdot \left(x \cdot {\varepsilon}^2\right)\right)\]
      9.0
    3. Taylor expanded around 0 to get
      \[\color{red}{\varepsilon - \left(\frac{1}{2} \cdot \left({x}^2 \cdot \varepsilon\right) + \frac{1}{2} \cdot \left(x \cdot {\varepsilon}^2\right)\right)} \leadsto \color{blue}{\varepsilon - \left(\frac{1}{2} \cdot \left({x}^2 \cdot \varepsilon\right) + \frac{1}{2} \cdot \left(x \cdot {\varepsilon}^2\right)\right)}\]
      9.0
    4. Applied simplify to get
      \[\color{red}{\varepsilon - \left(\frac{1}{2} \cdot \left({x}^2 \cdot \varepsilon\right) + \frac{1}{2} \cdot \left(x \cdot {\varepsilon}^2\right)\right)} \leadsto \color{blue}{\varepsilon - \left(\left(x + \varepsilon\right) \cdot \left(x \cdot \varepsilon\right)\right) \cdot \frac{1}{2}}\]
      0.2

    if 2.0116360245016707e-26 < eps

    1. Started with
      \[\sin \left(x + \varepsilon\right) - \sin x\]
      29.7
    2. Using strategy rm
      29.7
    3. Applied sin-sum to get
      \[\color{red}{\sin \left(x + \varepsilon\right)} - \sin x \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
      2.1
    4. Applied associate--l+ to get
      \[\color{red}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x} \leadsto \color{blue}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)}\]
      2.1
    5. Using strategy rm
      2.1
    6. Applied flip-- to get
      \[\sin x \cdot \cos \varepsilon + \color{red}{\left(\cos x \cdot \sin \varepsilon - \sin x\right)} \leadsto \sin x \cdot \cos \varepsilon + \color{blue}{\frac{{\left(\cos x \cdot \sin \varepsilon\right)}^2 - {\left(\sin x\right)}^2}{\cos x \cdot \sin \varepsilon + \sin x}}\]
      2.2
  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)))))