\[\cos \left(x + \varepsilon\right) - \cos x\]

⬇

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -8.658073314327725 \cdot 10^{-05}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sqrt[3]{{\left(\sin x\right)}^3 \cdot {\left(\sin \varepsilon\right)}^3}\right) - \cos x\\ \mathbf{if}\;\varepsilon \le -2.3361824468930465 \cdot 10^{-158}:\\ \;\;\;\;-\left(\frac{1}{2} \cdot {\varepsilon}^2 + \sqrt[3]{{\left(\sin x\right)}^3 \cdot {\left(\sin \varepsilon\right)}^3}\right)\\ \mathbf{if}\;\varepsilon \le 4.1153145131375503 \cdot 10^{-94}:\\ \;\;\;\;\left(\varepsilon \cdot \frac{1}{6}\right) \cdot {x}^3 - \varepsilon \cdot \left(\frac{1}{2} \cdot \varepsilon + x\right)\\ \mathbf{if}\;\varepsilon \le 3.384489567637978 \cdot 10^{-06}:\\ \;\;\;\;-\left(\frac{1}{2} \cdot {\varepsilon}^2 + \sqrt[3]{{\left(\sin x\right)}^3 \cdot {\left(\sin \varepsilon\right)}^3}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sqrt[3]{{\left(\sin x\right)}^3 \cdot {\left(\sin \varepsilon\right)}^3}\right) - \cos x\\ \end{array}\]

Derivation

Split input into 3 regimes.
if eps < -8.658073314327725e-05 or 3.384489567637978e-06 < eps
1. Initial program 30.2
  \[\cos \left(x + \varepsilon\right) - \cos x\]
2. Using strategy rm
3. Applied cos-sum 1.0
  \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
4. Using strategy rm
5. Applied add-cbrt-cube 1.0
  \[\leadsto \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \color{blue}{\sqrt[3]{{\left(\sin \varepsilon\right)}^3}}\right) - \cos x\]
6. Applied add-cbrt-cube 1.1
  \[\leadsto \left(\cos x \cdot \cos \varepsilon - \color{blue}{\sqrt[3]{{\left(\sin x\right)}^3}} \cdot \sqrt[3]{{\left(\sin \varepsilon\right)}^3}\right) - \cos x\]
7. Applied cbrt-unprod 1.1
  \[\leadsto \left(\cos x \cdot \cos \varepsilon - \color{blue}{\sqrt[3]{{\left(\sin x\right)}^3 \cdot {\left(\sin \varepsilon\right)}^3}}\right) - \cos x\]
if -8.658073314327725e-05 < eps < -2.3361824468930465e-158 or 4.1153145131375503e-94 < eps < 3.384489567637978e-06
1. Initial program 60.9
  \[\cos \left(x + \varepsilon\right) - \cos x\]
2. Using strategy rm
3. Applied cos-sum 59.7
  \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
4. Using strategy rm
5. Applied add-cbrt-cube 59.7
  \[\leadsto \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \color{blue}{\sqrt[3]{{\left(\sin \varepsilon\right)}^3}}\right) - \cos x\]
6. Applied add-cbrt-cube 59.7
  \[\leadsto \left(\cos x \cdot \cos \varepsilon - \color{blue}{\sqrt[3]{{\left(\sin x\right)}^3}} \cdot \sqrt[3]{{\left(\sin \varepsilon\right)}^3}\right) - \cos x\]
7. Applied cbrt-unprod 59.7
  \[\leadsto \left(\cos x \cdot \cos \varepsilon - \color{blue}{\sqrt[3]{{\left(\sin x\right)}^3 \cdot {\left(\sin \varepsilon\right)}^3}}\right) - \cos x\]
8. Applied taylor 11.5
  \[\leadsto -\left(\frac{1}{2} \cdot {\varepsilon}^2 + \sqrt[3]{{\left(\sin x\right)}^3 \cdot {\left(\sin \varepsilon\right)}^3}\right)\]
9. Taylor expanded around 0 11.5
  
  \[\leadsto \color{blue}{-\left(\frac{1}{2} \cdot {\varepsilon}^2 + \sqrt[3]{{\left(\sin x\right)}^3 \cdot {\left(\sin \varepsilon\right)}^3}\right)}\]
if -2.3361824468930465e-158 < eps < 4.1153145131375503e-94
1. Initial program 38.2
  \[\cos \left(x + \varepsilon\right) - \cos x\]
2. Applied taylor 9.5
  \[\leadsto \frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)\]
3. Taylor expanded around 0 9.5
  
  \[\leadsto \color{blue}{\frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)}\]
4. Applied simplify 9.5
  \[\leadsto \color{blue}{\left(\varepsilon \cdot \frac{1}{6}\right) \cdot {x}^3 - \varepsilon \cdot \left(\frac{1}{2} \cdot \varepsilon + x\right)}\]
Recombined 3 regimes into one program.
Removed slow pow expressions

Error

Derivation

`if eps < -8.658073314327725e-05 or 3.384489567637978e-06 < eps`

`if -8.658073314327725e-05 < eps < -2.3361824468930465e-158 or 4.1153145131375503e-94 < eps < 3.384489567637978e-06`

`if -2.3361824468930465e-158 < eps < 4.1153145131375503e-94`

Runtime