Average Error: 15.3 → 0.6
Time: 6.7s
Precision: 64
Internal Precision: 1344
$1 - \cos \left(\frac{x}{2}\right)$
$\frac{\sin \left(\frac{x}{2}\right) \cdot \sin \left(\frac{x}{2}\right)}{e^{\log \left(1 + \cos \left(\frac{x}{2}\right)\right)}}$

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# Derivation

1. Initial program 15.3

$1 - \cos \left(\frac{x}{2}\right)$
2. Initial simplification15.3

$\leadsto 1 - \cos \left(\frac{x}{2}\right)$
3. Using strategy rm

$\leadsto \color{blue}{e^{\log \left(1 - \cos \left(\frac{x}{2}\right)\right)}}$
5. Using strategy rm
6. Applied flip--15.6

$\leadsto e^{\log \color{blue}{\left(\frac{1 \cdot 1 - \cos \left(\frac{x}{2}\right) \cdot \cos \left(\frac{x}{2}\right)}{1 + \cos \left(\frac{x}{2}\right)}\right)}}$
7. Applied log-div15.6

$\leadsto e^{\color{blue}{\log \left(1 \cdot 1 - \cos \left(\frac{x}{2}\right) \cdot \cos \left(\frac{x}{2}\right)\right) - \log \left(1 + \cos \left(\frac{x}{2}\right)\right)}}$
8. Applied exp-diff15.6

$\leadsto \color{blue}{\frac{e^{\log \left(1 \cdot 1 - \cos \left(\frac{x}{2}\right) \cdot \cos \left(\frac{x}{2}\right)\right)}}{e^{\log \left(1 + \cos \left(\frac{x}{2}\right)\right)}}}$
9. Simplified0.6

$\leadsto \frac{\color{blue}{\sin \left(\frac{x}{2}\right) \cdot \sin \left(\frac{x}{2}\right)}}{e^{\log \left(1 + \cos \left(\frac{x}{2}\right)\right)}}$
10. Final simplification0.6

$\leadsto \frac{\sin \left(\frac{x}{2}\right) \cdot \sin \left(\frac{x}{2}\right)}{e^{\log \left(1 + \cos \left(\frac{x}{2}\right)\right)}}$

# Runtime

Time bar (total: 6.7s)Debug log

herbie shell --seed '#(2775764126 3555076145 3898259844 1891440260 2599947619 1948460636)'
(FPCore (x)
:name "1 - cos(x / 2)"
(- 1 (cos (/ x 2))))