\[\sin \left(x + \varepsilon\right) - \sin x\]
Test:
NMSE example 3.3
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 12.8 s
Input Error: 16.9
Output Error: 0.3
Log: ⚲
Profile: 🕒
\(\begin{cases} \sqrt[3]{{\left(\sin \varepsilon\right)}^3 \cdot {\left(\cos x\right)}^3} - \left(\sin x - \cos \varepsilon \cdot \sin x\right) & \text{when } \varepsilon \le -0.027100457f0 \\ \left(\varepsilon - \frac{1}{6} \cdot {\varepsilon}^3\right) \cdot \cos x - \left(\sin x \cdot \frac{1}{2}\right) \cdot {\varepsilon}^2 & \text{when } \varepsilon \le 0.00019073929f0 \\ \sqrt[3]{{\left(\sin \varepsilon\right)}^3 \cdot {\left(\cos x\right)}^3} - \left(\sin x - \cos \varepsilon \cdot \sin x\right) & \text{otherwise} \end{cases}\)

    if eps < -0.027100457f0 or 0.00019073929f0 < 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
      \[\color{red}{\sin \left(x + \varepsilon\right)} - \sin x \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
      0.4
    4. Applied associate--l+ to get
      \[\color{red}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x} \leadsto \color{blue}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)}\]
      0.4
    5. Using strategy rm
      0.4
    6. Applied add-cbrt-cube to get
      \[\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \color{red}{\sin \varepsilon} - \sin x\right) \leadsto \sin x \cdot \cos \varepsilon + \left(\cos x \cdot \color{blue}{\sqrt[3]{{\left(\sin \varepsilon\right)}^3}} - \sin x\right)\]
      0.4
    7. Applied add-cbrt-cube to get
      \[\sin x \cdot \cos \varepsilon + \left(\color{red}{\cos x} \cdot \sqrt[3]{{\left(\sin \varepsilon\right)}^3} - \sin x\right) \leadsto \sin x \cdot \cos \varepsilon + \left(\color{blue}{\sqrt[3]{{\left(\cos x\right)}^3}} \cdot \sqrt[3]{{\left(\sin \varepsilon\right)}^3} - \sin x\right)\]
      0.4
    8. Applied cbrt-unprod to get
      \[\sin x \cdot \cos \varepsilon + \left(\color{red}{\sqrt[3]{{\left(\cos x\right)}^3} \cdot \sqrt[3]{{\left(\sin \varepsilon\right)}^3}} - \sin x\right) \leadsto \sin x \cdot \cos \varepsilon + \left(\color{blue}{\sqrt[3]{{\left(\cos x\right)}^3 \cdot {\left(\sin \varepsilon\right)}^3}} - \sin x\right)\]
      0.5
    9. Applied taylor to get
      \[\sin x \cdot \cos \varepsilon + \left(\sqrt[3]{{\left(\cos x\right)}^3 \cdot {\left(\sin \varepsilon\right)}^3} - \sin x\right) \leadsto \sin x \cdot \cos \varepsilon + \left(\sqrt[3]{{\left(\cos x\right)}^3 \cdot {\left(\sin \varepsilon\right)}^3} - \sin x\right)\]
      0.5
    10. Taylor expanded around 0 to get
      \[\sin x \cdot \cos \varepsilon + \left(\color{red}{\sqrt[3]{{\left(\cos x\right)}^3 \cdot {\left(\sin \varepsilon\right)}^3}} - \sin x\right) \leadsto \sin x \cdot \cos \varepsilon + \left(\color{blue}{\sqrt[3]{{\left(\cos x\right)}^3 \cdot {\left(\sin \varepsilon\right)}^3}} - \sin x\right)\]
      0.5
    11. Applied simplify to get
      \[\sin x \cdot \cos \varepsilon + \left(\sqrt[3]{{\left(\cos x\right)}^3 \cdot {\left(\sin \varepsilon\right)}^3} - \sin x\right) \leadsto \sqrt[3]{{\left(\sin \varepsilon\right)}^3 \cdot {\left(\cos x\right)}^3} - \left(\sin x - \cos \varepsilon \cdot \sin x\right)\]
      0.5
    12. Applied final simplification

    if -0.027100457f0 < eps < 0.00019073929f0

    1. Started with
      \[\sin \left(x + \varepsilon\right) - \sin x\]
      20.0
    2. Using strategy rm
      20.0
    3. Applied sin-sum to get
      \[\color{red}{\sin \left(x + \varepsilon\right)} - \sin x \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
      13.2
    4. Applied associate--l+ to get
      \[\color{red}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x} \leadsto \color{blue}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)}\]
      13.2
    5. Using strategy rm
      13.2
    6. Applied add-cbrt-cube to get
      \[\color{red}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)} \leadsto \color{blue}{\sqrt[3]{{\left(\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\right)}^3}}\]
      19.5
    7. Applied taylor to get
      \[\sqrt[3]{{\left(\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\right)}^3} \leadsto \varepsilon \cdot \cos x - \left(\frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + \frac{1}{2} \cdot \left({\varepsilon}^2 \cdot \sin x\right)\right)\]
      0.1
    8. 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
    9. 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
    10. 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)))))