Average Error: 30.9 → 0.3
Time: 33.2s
Precision: 64
Internal Precision: 2368
$\frac{1 - \cos x}{x \cdot x}$
$\begin{array}{l} \mathbf{if}\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2} \le 0.5141478801211188:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array}$

# Try it out

Results

 In Out
Enter valid numbers for all inputs

# Derivation

1. Split input into 2 regimes
2. ## if (- (+ (* 1/720 (pow x 4)) 1/2) (* 1/24 (pow x 2))) < 0.5141478801211188

1. Initial program 60.9

$\frac{1 - \cos x}{x \cdot x}$
2. Taylor expanded around 0 0.2

$\leadsto \color{blue}{\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}}$

## if 0.5141478801211188 < (- (+ (* 1/720 (pow x 4)) 1/2) (* 1/24 (pow x 2)))

1. Initial program 1.0

$\frac{1 - \cos x}{x \cdot x}$
2. Using strategy rm
3. Applied associate-/r*0.5

$\leadsto \color{blue}{\frac{\frac{1 - \cos x}{x}}{x}}$
3. Recombined 2 regimes into one program.

# Runtime

Time bar (total: 33.2s)Debug log

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