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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -0.00041039810197564524:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \le 0.00010508423175312767:\\ \;\;\;\;\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \log_* (1 + (e^{\sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)} - 1)^*)\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \end{array}\]

Derivation

Split input into 2 regimes
if eps < -0.00041039810197564524 or 0.00010508423175312767 < eps
1. Initial program 30.2
  \[\cos \left(x + \varepsilon\right) - \cos x\]
2. Using strategy rm
3. Applied cos-sum0.9
  \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
4. Applied associate--l-0.9
  \[\leadsto \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]
if -0.00041039810197564524 < eps < 0.00010508423175312767
1. Initial program 48.9
  \[\cos \left(x + \varepsilon\right) - \cos x\]
2. Using strategy rm
3. Applied diff-cos37.2
  \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)}\]
4. Simplified0.5
  \[\leadsto -2 \cdot \color{blue}{\left(\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)}\]
5. Using strategy rm
6. Applied log1p-expm1-u0.6
  \[\leadsto -2 \cdot \left(\color{blue}{\log_* (1 + (e^{\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)} - 1)^*)} \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]
Recombined 2 regimes into one program.
Final simplification0.8
\[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -0.00041039810197564524:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \le 0.00010508423175312767:\\ \;\;\;\;\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \log_* (1 + (e^{\sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)} - 1)^*)\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \end{array}\]

Error

Derivation

`if eps < -0.00041039810197564524 or 0.00010508423175312767 < eps`

`if -0.00041039810197564524 < eps < 0.00010508423175312767`

Reproduce