Average Error: 32.8 → 0.2
Time: 55.7s
Precision: 64
Internal Precision: 4416
$\frac{\left(2 \cdot a - 3 \cdot \sin a\right) + a \cdot \cos a}{\left(\left(\left(\left(2 \cdot a\right) \cdot a\right) \cdot a\right) \cdot a\right) \cdot a}$
$\begin{array}{l} \mathbf{if}\;a \le -0.14258269255488928:\\ \;\;\;\;\frac{1}{a \cdot a} \cdot \frac{\left({\left(\cos a\right)}^{3} + {2}^{3}\right) \cdot \frac{a}{3} - \sin a \cdot \left(\cos a \cdot \cos a + \left(2 \cdot 2 - 2 \cdot \cos a\right)\right)}{\left(\left(2 \cdot a\right) \cdot a\right) \cdot \left(\left(\cos a \cdot \cos a + \left(2 \cdot 2 - 2 \cdot \cos a\right)\right) \cdot \frac{a}{3}\right)}\\ \mathbf{elif}\;a \le 0.14139226800334523:\\ \;\;\;\;\left(\frac{1}{120} + {a}^{4} \cdot \frac{1}{120960}\right) - {a}^{2} \cdot \frac{1}{2520}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos a + 2}{\left(2 \cdot a\right) \cdot {a}^{3}} - \frac{\sin a \cdot 3}{\left(a \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot \left(2 \cdot a\right)\right)}\\ \end{array}$

# Try it out

Results

 In Out
Enter valid numbers for all inputs

# Derivation

1. Split input into 3 regimes
2. ## if a < -0.14258269255488928

1. Initial program 3.7

$\frac{\left(2 \cdot a - 3 \cdot \sin a\right) + a \cdot \cos a}{\left(\left(\left(\left(2 \cdot a\right) \cdot a\right) \cdot a\right) \cdot a\right) \cdot a}$
2. Initial simplification3.7

$\leadsto \frac{\left(\cos a + 2\right) \cdot a - 3 \cdot \sin a}{\left(a \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot \left(2 \cdot a\right)\right)}$
3. Using strategy rm
4. Applied *-un-lft-identity3.7

$\leadsto \frac{\color{blue}{1 \cdot \left(\left(\cos a + 2\right) \cdot a - 3 \cdot \sin a\right)}}{\left(a \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot \left(2 \cdot a\right)\right)}$
5. Applied times-frac0.3

$\leadsto \color{blue}{\frac{1}{a \cdot a} \cdot \frac{\left(\cos a + 2\right) \cdot a - 3 \cdot \sin a}{\left(a \cdot a\right) \cdot \left(2 \cdot a\right)}}$
6. Simplified0.2

$\leadsto \frac{1}{a \cdot a} \cdot \color{blue}{\frac{\left(\cos a + 2\right) - \frac{\sin a}{\frac{a}{3}}}{\left(a \cdot 2\right) \cdot a}}$
7. Using strategy rm
8. Applied flip3-+0.2

$\leadsto \frac{1}{a \cdot a} \cdot \frac{\color{blue}{\frac{{\left(\cos a\right)}^{3} + {2}^{3}}{\cos a \cdot \cos a + \left(2 \cdot 2 - \cos a \cdot 2\right)}} - \frac{\sin a}{\frac{a}{3}}}{\left(a \cdot 2\right) \cdot a}$
9. Applied frac-sub0.3

$\leadsto \frac{1}{a \cdot a} \cdot \frac{\color{blue}{\frac{\left({\left(\cos a\right)}^{3} + {2}^{3}\right) \cdot \frac{a}{3} - \left(\cos a \cdot \cos a + \left(2 \cdot 2 - \cos a \cdot 2\right)\right) \cdot \sin a}{\left(\cos a \cdot \cos a + \left(2 \cdot 2 - \cos a \cdot 2\right)\right) \cdot \frac{a}{3}}}}{\left(a \cdot 2\right) \cdot a}$
10. Applied associate-/l/0.3

$\leadsto \frac{1}{a \cdot a} \cdot \color{blue}{\frac{\left({\left(\cos a\right)}^{3} + {2}^{3}\right) \cdot \frac{a}{3} - \left(\cos a \cdot \cos a + \left(2 \cdot 2 - \cos a \cdot 2\right)\right) \cdot \sin a}{\left(\left(a \cdot 2\right) \cdot a\right) \cdot \left(\left(\cos a \cdot \cos a + \left(2 \cdot 2 - \cos a \cdot 2\right)\right) \cdot \frac{a}{3}\right)}}$

## if -0.14258269255488928 < a < 0.14139226800334523

1. Initial program 62.1

$\frac{\left(2 \cdot a - 3 \cdot \sin a\right) + a \cdot \cos a}{\left(\left(\left(\left(2 \cdot a\right) \cdot a\right) \cdot a\right) \cdot a\right) \cdot a}$
2. Initial simplification61.8

$\leadsto \frac{\left(\cos a + 2\right) \cdot a - 3 \cdot \sin a}{\left(a \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot \left(2 \cdot a\right)\right)}$
3. Taylor expanded around 0 0.0

$\leadsto \color{blue}{\left(\frac{1}{120} + \frac{1}{120960} \cdot {a}^{4}\right) - \frac{1}{2520} \cdot {a}^{2}}$

## if 0.14139226800334523 < a

1. Initial program 3.8

$\frac{\left(2 \cdot a - 3 \cdot \sin a\right) + a \cdot \cos a}{\left(\left(\left(\left(2 \cdot a\right) \cdot a\right) \cdot a\right) \cdot a\right) \cdot a}$
2. Initial simplification3.8

$\leadsto \frac{\left(\cos a + 2\right) \cdot a - 3 \cdot \sin a}{\left(a \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot \left(2 \cdot a\right)\right)}$
3. Using strategy rm
4. Applied div-sub3.9

$\leadsto \color{blue}{\frac{\left(\cos a + 2\right) \cdot a}{\left(a \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot \left(2 \cdot a\right)\right)} - \frac{3 \cdot \sin a}{\left(a \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot \left(2 \cdot a\right)\right)}}$
5. Simplified0.3

$\leadsto \color{blue}{\frac{\cos a + 2}{\left(a \cdot 2\right) \cdot {a}^{3}}} - \frac{3 \cdot \sin a}{\left(a \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot \left(2 \cdot a\right)\right)}$
3. Recombined 3 regimes into one program.
4. Final simplification0.2

$\leadsto \begin{array}{l} \mathbf{if}\;a \le -0.14258269255488928:\\ \;\;\;\;\frac{1}{a \cdot a} \cdot \frac{\left({\left(\cos a\right)}^{3} + {2}^{3}\right) \cdot \frac{a}{3} - \sin a \cdot \left(\cos a \cdot \cos a + \left(2 \cdot 2 - 2 \cdot \cos a\right)\right)}{\left(\left(2 \cdot a\right) \cdot a\right) \cdot \left(\left(\cos a \cdot \cos a + \left(2 \cdot 2 - 2 \cdot \cos a\right)\right) \cdot \frac{a}{3}\right)}\\ \mathbf{elif}\;a \le 0.14139226800334523:\\ \;\;\;\;\left(\frac{1}{120} + {a}^{4} \cdot \frac{1}{120960}\right) - {a}^{2} \cdot \frac{1}{2520}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos a + 2}{\left(2 \cdot a\right) \cdot {a}^{3}} - \frac{\sin a \cdot 3}{\left(a \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot \left(2 \cdot a\right)\right)}\\ \end{array}$

# Runtime

Time bar (total: 55.7s)Debug log

herbie shell --seed '#(2775764126 3555076145 3898259844 1891440260 2599947619 1948460636)'
(FPCore (a)
:name "(2*a - 3*sin(a) + a*cos(a))/(2*a*a*a*a*a)"
(/ (+ (- (* 2 a) (* 3 (sin a))) (* a (cos a))) (* (* (* (* (* 2 a) a) a) a) a)))