Average Error: 35.8 → 0.4
Time: 17.6s
Precision: 64
Internal Precision: 2368
$\frac{a - \sin a}{\left(a \cdot a\right) \cdot a}$
$\begin{array}{l} \mathbf{if}\;a \le -0.043473338976223706:\\ \;\;\;\;\frac{1}{a \cdot a} \cdot \frac{a - \sin a}{a}\\ \mathbf{elif}\;a \le 0.046591036094757086:\\ \;\;\;\;\left(\frac{1}{5040} \cdot {a}^{4} + \frac{1}{6}\right) - \frac{1}{120} \cdot {a}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt[3]{a - \sin a}}{a} \cdot \frac{\sqrt[3]{a - \sin a}}{a}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{a - \sin a} \cdot \sqrt[3]{a - \sin a}} \cdot \sqrt[3]{\sqrt[3]{a - \sin a}}}{a}\\ \end{array}$

# Try it out

Results

 In Out
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# Derivation

1. Split input into 3 regimes
2. ## if a < -0.043473338976223706

1. Initial program 10.3

$\frac{a - \sin a}{\left(a \cdot a\right) \cdot a}$
2. Initial simplification10.3

$\leadsto \frac{a - \sin a}{{a}^{3}}$
3. Using strategy rm
4. Applied unpow310.3

$\leadsto \frac{a - \sin a}{\color{blue}{\left(a \cdot a\right) \cdot a}}$
5. Applied *-un-lft-identity10.3

$\leadsto \frac{\color{blue}{1 \cdot \left(a - \sin a\right)}}{\left(a \cdot a\right) \cdot a}$
6. Applied times-frac0.7

$\leadsto \color{blue}{\frac{1}{a \cdot a} \cdot \frac{a - \sin a}{a}}$

## if -0.043473338976223706 < a < 0.046591036094757086

1. Initial program 61.2

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

$\leadsto \frac{a - \sin a}{{a}^{3}}$
3. Taylor expanded around 0 0.0

$\leadsto \color{blue}{\left(\frac{1}{6} + \frac{1}{5040} \cdot {a}^{4}\right) - \frac{1}{120} \cdot {a}^{2}}$

## if 0.046591036094757086 < a

1. Initial program 11.0

$\frac{a - \sin a}{\left(a \cdot a\right) \cdot a}$
2. Initial simplification11.0

$\leadsto \frac{a - \sin a}{{a}^{3}}$
3. Using strategy rm

$\leadsto \frac{a - \sin a}{\color{blue}{\left(\sqrt[3]{{a}^{3}} \cdot \sqrt[3]{{a}^{3}}\right) \cdot \sqrt[3]{{a}^{3}}}}$

$\leadsto \frac{\color{blue}{\left(\sqrt[3]{a - \sin a} \cdot \sqrt[3]{a - \sin a}\right) \cdot \sqrt[3]{a - \sin a}}}{\left(\sqrt[3]{{a}^{3}} \cdot \sqrt[3]{{a}^{3}}\right) \cdot \sqrt[3]{{a}^{3}}}$
6. Applied times-frac11.4

$\leadsto \color{blue}{\frac{\sqrt[3]{a - \sin a} \cdot \sqrt[3]{a - \sin a}}{\sqrt[3]{{a}^{3}} \cdot \sqrt[3]{{a}^{3}}} \cdot \frac{\sqrt[3]{a - \sin a}}{\sqrt[3]{{a}^{3}}}}$
7. Simplified11.3

$\leadsto \color{blue}{\left(\frac{\sqrt[3]{a - \sin a}}{a} \cdot \frac{\sqrt[3]{a - \sin a}}{a}\right)} \cdot \frac{\sqrt[3]{a - \sin a}}{\sqrt[3]{{a}^{3}}}$
8. Simplified0.7

$\leadsto \left(\frac{\sqrt[3]{a - \sin a}}{a} \cdot \frac{\sqrt[3]{a - \sin a}}{a}\right) \cdot \color{blue}{\frac{\sqrt[3]{a - \sin a}}{a}}$
9. Using strategy rm

$\leadsto \left(\frac{\sqrt[3]{a - \sin a}}{a} \cdot \frac{\sqrt[3]{a - \sin a}}{a}\right) \cdot \frac{\sqrt[3]{\color{blue}{\left(\sqrt[3]{a - \sin a} \cdot \sqrt[3]{a - \sin a}\right) \cdot \sqrt[3]{a - \sin a}}}}{a}$
11. Applied cbrt-prod0.7

$\leadsto \left(\frac{\sqrt[3]{a - \sin a}}{a} \cdot \frac{\sqrt[3]{a - \sin a}}{a}\right) \cdot \frac{\color{blue}{\sqrt[3]{\sqrt[3]{a - \sin a} \cdot \sqrt[3]{a - \sin a}} \cdot \sqrt[3]{\sqrt[3]{a - \sin a}}}}{a}$
3. Recombined 3 regimes into one program.
4. Final simplification0.4

$\leadsto \begin{array}{l} \mathbf{if}\;a \le -0.043473338976223706:\\ \;\;\;\;\frac{1}{a \cdot a} \cdot \frac{a - \sin a}{a}\\ \mathbf{elif}\;a \le 0.046591036094757086:\\ \;\;\;\;\left(\frac{1}{5040} \cdot {a}^{4} + \frac{1}{6}\right) - \frac{1}{120} \cdot {a}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt[3]{a - \sin a}}{a} \cdot \frac{\sqrt[3]{a - \sin a}}{a}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{a - \sin a} \cdot \sqrt[3]{a - \sin a}} \cdot \sqrt[3]{\sqrt[3]{a - \sin a}}}{a}\\ \end{array}$

# Runtime

Time bar (total: 17.6s)Debug log

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