Average Error: 39.8 → 0.7
Time: 15.3s
Precision: 64
Internal Precision: 2368
$\cos \left(x + e\right) - \cos x$
$\begin{array}{l} \mathbf{if}\;e \le -0.00021352682501638267:\\ \;\;\;\;\cos x \cdot \cos e - \left(\cos x + \sin e \cdot \sin x\right)\\ \mathbf{elif}\;e \le 1.1702967621355792 \cdot 10^{-06}:\\ \;\;\;\;\left(\sin \left(\frac{e}{2}\right) \cdot \sin \left(\frac{\left(x + e\right) + x}{2}\right)\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos e - \sin e \cdot \sin x\right) - \cos x\\ \end{array}$

# Try it out

Results

 In Out
Enter valid numbers for all inputs

# Derivation

1. Split input into 3 regimes
2. ## if e < -0.00021352682501638267

1. Initial program 29.9

$\cos \left(x + e\right) - \cos x$
2. Using strategy rm
3. Applied cos-sum0.9

$\leadsto \color{blue}{\left(\cos x \cdot \cos e - \sin x \cdot \sin e\right)} - \cos x$
4. Applied associate--l-0.9

$\leadsto \color{blue}{\cos x \cdot \cos e - \left(\sin x \cdot \sin e + \cos x\right)}$

## if -0.00021352682501638267 < e < 1.1702967621355792e-06

1. Initial program 49.4

$\cos \left(x + e\right) - \cos x$
2. Using strategy rm
3. Applied diff-cos38.0

$\leadsto \color{blue}{-2 \cdot \left(\sin \left(\frac{\left(x + e\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + e\right) + x}{2}\right)\right)}$
4. Simplified0.5

$\leadsto -2 \cdot \color{blue}{\left(\sin \left(\frac{x + \left(e + x\right)}{2}\right) \cdot \sin \left(\frac{e}{2}\right)\right)}$

## if 1.1702967621355792e-06 < e

1. Initial program 30.4

$\cos \left(x + e\right) - \cos x$
2. Using strategy rm
3. Applied cos-sum1.0

$\leadsto \color{blue}{\left(\cos x \cdot \cos e - \sin x \cdot \sin e\right)} - \cos x$
3. Recombined 3 regimes into one program.
4. Final simplification0.7

$\leadsto \begin{array}{l} \mathbf{if}\;e \le -0.00021352682501638267:\\ \;\;\;\;\cos x \cdot \cos e - \left(\cos x + \sin e \cdot \sin x\right)\\ \mathbf{elif}\;e \le 1.1702967621355792 \cdot 10^{-06}:\\ \;\;\;\;\left(\sin \left(\frac{e}{2}\right) \cdot \sin \left(\frac{\left(x + e\right) + x}{2}\right)\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos e - \sin e \cdot \sin x\right) - \cos x\\ \end{array}$

# Runtime

Time bar (total: 15.3s)Debug log

herbie shell --seed '#(2775764126 3555076145 3898259844 1891440260 2599947619 1948460636)'
(FPCore (x e)
:name "cos(x+e)-cos(x)"
(- (cos (+ x e)) (cos x)))