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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -0.0009727920975067085:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\\ \mathbf{if}\;\varepsilon \le 7.590472496670245 \cdot 10^{-09}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

Derivation

Split input into 3 regimes
if eps < -0.0009727920975067085
1. Initial program 29.9
  \[\cos \left(x + \varepsilon\right) - \cos x\]
2. Using strategy rm
3. Applied cos-sum0.8
  \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
4. Applied associate--l-0.9
  \[\leadsto \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]
if -0.0009727920975067085 < eps < 7.590472496670245e-09
1. Initial program 49.4
  \[\cos \left(x + \varepsilon\right) - \cos x\]
2. Using strategy rm
3. Applied diff-cos37.4
  \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)}\]
4. Applied simplify0.5
  \[\leadsto -2 \cdot \color{blue}{\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)\right)}\]
if 7.590472496670245e-09 < eps
1. Initial program 30.4
  \[\cos \left(x + \varepsilon\right) - \cos x\]
2. Using strategy rm
3. Applied cos-sum1.2
  \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
Recombined 3 regimes into one program.
Removed slow pow expressions.

Runtime

herbie shell --seed '#(1063185673 2139736501 2393378123 1907444849 1070993796 1007244912)' 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))

Error

Derivation

`if eps < -0.0009727920975067085`

`if -0.0009727920975067085 < eps < 7.590472496670245e-09`

`if 7.590472496670245e-09 < eps`

Runtime