\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.9972861708012687 \lor \neg \left(x \le 107.4189493580581\right):\\ \;\;\;\;\frac{\frac{2}{x}}{x \cdot x} + \left(\frac{2}{{x}^{7}} + \frac{2}{{x}^{5}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x - 1} + \left(\sqrt{\frac{1}{1 + x}} \cdot \sqrt{\frac{1}{1 + x}} - \frac{2}{x}\right)\\ \end{array}\]

Original	9.9
Target	0.3
Herbie	0.1

Derivation

Split input into 2 regimes
if x < -0.9972861708012687 or 107.4189493580581 < x
1. Initial program 20.0
  \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
2. Taylor expanded around inf 0.7
  \[\leadsto \color{blue}{2 \cdot \frac{1}{{x}^{7}} + \left(2 \cdot \frac{1}{{x}^{3}} + 2 \cdot \frac{1}{{x}^{5}}\right)}\]
3. Simplified0.2
  \[\leadsto \color{blue}{\left(\frac{2}{{x}^{7}} + \frac{2}{{x}^{5}}\right) + \frac{\frac{2}{x}}{x \cdot x}}\]
if -0.9972861708012687 < x < 107.4189493580581
1. Initial program 0.0
  \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
2. Using strategy rm
3. Applied add-sqr-sqrt0.0
  \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{x + 1}} \cdot \sqrt{\frac{1}{x + 1}}} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
Recombined 2 regimes into one program.
Final simplification0.1
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.9972861708012687 \lor \neg \left(x \le 107.4189493580581\right):\\ \;\;\;\;\frac{\frac{2}{x}}{x \cdot x} + \left(\frac{2}{{x}^{7}} + \frac{2}{{x}^{5}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x - 1} + \left(\sqrt{\frac{1}{1 + x}} \cdot \sqrt{\frac{1}{1 + x}} - \frac{2}{x}\right)\\ \end{array}\]

Runtime

herbie shell --seed 2018251 
(FPCore (x)
  :name "3frac (problem 3.3.3)"

  :herbie-target
  (/ 2 (* x (- (* x x) 1)))

  (+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1))))

Error

Try it out

Target

Derivation

`if x < -0.9972861708012687 or 107.4189493580581 < x`

`if -0.9972861708012687 < x < 107.4189493580581`

Runtime

In
Out