Average Error: 37.0 → 14.8
Time: 47.9s
Precision: 64
Internal Precision: 2368
$\tan \left(x + e\right) - \tan x$
$\begin{array}{l} \mathbf{if}\;x \cdot {e}^{2} + \left({e}^{3} \cdot {x}^{2} + e\right) \le -6.095429746541726 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{{\left(\tan x\right)}^{3} + {\left(\tan e\right)}^{3}}{\tan x \cdot \tan x - \tan e \cdot \left(\tan x - \tan e\right)}}{1 - \tan x \cdot \tan e} - \tan x\\ \mathbf{if}\;x \cdot {e}^{2} + \left({e}^{3} \cdot {x}^{2} + e\right) \le 3.4462337190046724 \cdot 10^{-48}:\\ \;\;\;\;x \cdot {e}^{2} + \left({e}^{3} \cdot {x}^{2} + e\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan e}{\left(\sqrt[3]{1 - \tan x \cdot \tan e} \cdot \sqrt[3]{1 - \tan x \cdot \tan e}\right) \cdot \sqrt[3]{1 - \tan x \cdot \tan e}} - \tan x\\ \end{array}$

# Try it out

Results

 In Out
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# Derivation

1. Split input into 3 regimes
2. ## if (+ (* x (pow e 2)) (+ (* (pow e 3) (pow x 2)) e)) < -6.095429746541726e-25

1. Initial program 35.4

$\tan \left(x + e\right) - \tan x$
2. Using strategy rm
3. Applied tan-sum10.7

$\leadsto \color{blue}{\frac{\tan x + \tan e}{1 - \tan x \cdot \tan e}} - \tan x$
4. Using strategy rm
5. Applied flip3-+10.8

$\leadsto \frac{\color{blue}{\frac{{\left(\tan x\right)}^{3} + {\left(\tan e\right)}^{3}}{\tan x \cdot \tan x + \left(\tan e \cdot \tan e - \tan x \cdot \tan e\right)}}}{1 - \tan x \cdot \tan e} - \tan x$
6. Applied simplify10.8

$\leadsto \frac{\frac{{\left(\tan x\right)}^{3} + {\left(\tan e\right)}^{3}}{\color{blue}{\tan x \cdot \tan x - \tan e \cdot \left(\tan x - \tan e\right)}}}{1 - \tan x \cdot \tan e} - \tan x$

## if -6.095429746541726e-25 < (+ (* x (pow e 2)) (+ (* (pow e 3) (pow x 2)) e)) < 3.4462337190046724e-48

1. Initial program 40.1

$\tan \left(x + e\right) - \tan x$
2. Taylor expanded around 0 16.7

$\leadsto \color{blue}{x \cdot {e}^{2} + \left({e}^{3} \cdot {x}^{2} + e\right)}$

## if 3.4462337190046724e-48 < (+ (* x (pow e 2)) (+ (* (pow e 3) (pow x 2)) e))

1. Initial program 35.7

$\tan \left(x + e\right) - \tan x$
2. Using strategy rm
3. Applied tan-sum14.9

$\leadsto \color{blue}{\frac{\tan x + \tan e}{1 - \tan x \cdot \tan e}} - \tan x$
4. Using strategy rm
$\leadsto \frac{\tan x + \tan e}{\color{blue}{\left(\sqrt[3]{1 - \tan x \cdot \tan e} \cdot \sqrt[3]{1 - \tan x \cdot \tan e}\right) \cdot \sqrt[3]{1 - \tan x \cdot \tan e}}} - \tan x$
herbie shell --seed '#(2775764126 3555076145 3898259844 1891440260 2599947619 1948460636)'