\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
Test:
NMSE problem 3.3.3
Bits:
128 bits
Bits error versus x
Time: 10.5 s
Input Error: 9.5
Output Error: 0.3
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{(\left(-\left(1 + \frac{1}{x}\right)\right) * \left(-\left(\frac{1}{x} + (2 * \left(\frac{-1}{x}\right) + 2)_*\right)\right) + \left(\frac{1}{{x}^2} - \frac{1}{x}\right))_*}{\left(x - 1\right) \cdot (x * x + x)_*} & \text{when } x \le -1.5823654848886133 \\ \frac{\left(x - 1\right) \cdot \left(x - (2 * x + 2)_*\right) + \left(x \cdot x + x\right)}{\left(x - 1\right) \cdot (x * x + x)_*} & \text{when } x \le 40559.850189135206 \\ \frac{(\left(-\left(1 + \frac{1}{x}\right)\right) * \left(-\left(\frac{1}{x} + (2 * \left(\frac{-1}{x}\right) + 2)_*\right)\right) + \left(\frac{1}{{x}^2} - \frac{1}{x}\right))_*}{\left(x - 1\right) \cdot (x * x + x)_*} & \text{otherwise} \end{cases}\)

    if x < -1.5823654848886133 or 40559.850189135206 < x

    1. Started with
      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
      18.9
    2. Using strategy rm
      18.9
    3. Applied frac-sub to get
      \[\color{red}{\left(\frac{1}{x + 1} - \frac{2}{x}\right)} + \frac{1}{x - 1} \leadsto \color{blue}{\frac{1 \cdot x - \left(x + 1\right) \cdot 2}{\left(x + 1\right) \cdot x}} + \frac{1}{x - 1}\]
      52.4
    4. Applied frac-add to get
      \[\color{red}{\frac{1 \cdot x - \left(x + 1\right) \cdot 2}{\left(x + 1\right) \cdot x} + \frac{1}{x - 1}} \leadsto \color{blue}{\frac{\left(1 \cdot x - \left(x + 1\right) \cdot 2\right) \cdot \left(x - 1\right) + \left(\left(x + 1\right) \cdot x\right) \cdot 1}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}}\]
      50.3
    5. Applied simplify to get
      \[\frac{\color{red}{\left(1 \cdot x - \left(x + 1\right) \cdot 2\right) \cdot \left(x - 1\right) + \left(\left(x + 1\right) \cdot x\right) \cdot 1}}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)} \leadsto \frac{\color{blue}{(\left(x - 1\right) * \left(x - (2 * x + 2)_*\right) + \left(x \cdot x + x\right))_*}}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}\]
      51.0
    6. Applied simplify to get
      \[\frac{(\left(x - 1\right) * \left(x - (2 * x + 2)_*\right) + \left(x \cdot x + x\right))_*}{\color{red}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}} \leadsto \frac{(\left(x - 1\right) * \left(x - (2 * x + 2)_*\right) + \left(x \cdot x + x\right))_*}{\color{blue}{\left(x - 1\right) \cdot (x * x + x)_*}}\]
      51.0
    7. Applied taylor to get
      \[\frac{(\left(x - 1\right) * \left(x - (2 * x + 2)_*\right) + \left(x \cdot x + x\right))_*}{\left(x - 1\right) \cdot (x * x + x)_*} \leadsto \frac{(\left(-\left(1 + \frac{1}{x}\right)\right) * \left(-\left(\frac{1}{x} + (2 * \left(\frac{-1}{x}\right) + 2)_*\right)\right) + \left(\frac{1}{{x}^2} - \frac{1}{x}\right))_*}{\left(x - 1\right) \cdot (x * x + x)_*}\]
      0.6
    8. Taylor expanded around -inf to get
      \[\frac{\color{red}{(\left(-\left(1 + \frac{1}{x}\right)\right) * \left(-\left(\frac{1}{x} + (2 * \left(\frac{-1}{x}\right) + 2)_*\right)\right) + \left(\frac{1}{{x}^2} - \frac{1}{x}\right))_*}}{\left(x - 1\right) \cdot (x * x + x)_*} \leadsto \frac{\color{blue}{(\left(-\left(1 + \frac{1}{x}\right)\right) * \left(-\left(\frac{1}{x} + (2 * \left(\frac{-1}{x}\right) + 2)_*\right)\right) + \left(\frac{1}{{x}^2} - \frac{1}{x}\right))_*}}{\left(x - 1\right) \cdot (x * x + x)_*}\]
      0.6

    if -1.5823654848886133 < x < 40559.850189135206

    1. Started with
      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
      0.1
    2. Using strategy rm
      0.1
    3. Applied frac-sub to get
      \[\color{red}{\left(\frac{1}{x + 1} - \frac{2}{x}\right)} + \frac{1}{x - 1} \leadsto \color{blue}{\frac{1 \cdot x - \left(x + 1\right) \cdot 2}{\left(x + 1\right) \cdot x}} + \frac{1}{x - 1}\]
      0.1
    4. Applied frac-add to get
      \[\color{red}{\frac{1 \cdot x - \left(x + 1\right) \cdot 2}{\left(x + 1\right) \cdot x} + \frac{1}{x - 1}} \leadsto \color{blue}{\frac{\left(1 \cdot x - \left(x + 1\right) \cdot 2\right) \cdot \left(x - 1\right) + \left(\left(x + 1\right) \cdot x\right) \cdot 1}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}}\]
      0.0
    5. Applied simplify to get
      \[\frac{\color{red}{\left(1 \cdot x - \left(x + 1\right) \cdot 2\right) \cdot \left(x - 1\right) + \left(\left(x + 1\right) \cdot x\right) \cdot 1}}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)} \leadsto \frac{\color{blue}{(\left(x - 1\right) * \left(x - (2 * x + 2)_*\right) + \left(x \cdot x + x\right))_*}}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}\]
      0.1
    6. Applied simplify to get
      \[\frac{(\left(x - 1\right) * \left(x - (2 * x + 2)_*\right) + \left(x \cdot x + x\right))_*}{\color{red}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}} \leadsto \frac{(\left(x - 1\right) * \left(x - (2 * x + 2)_*\right) + \left(x \cdot x + x\right))_*}{\color{blue}{\left(x - 1\right) \cdot (x * x + x)_*}}\]
      0.1
    7. Using strategy rm
      0.1
    8. Applied fma-udef to get
      \[\frac{\color{red}{(\left(x - 1\right) * \left(x - (2 * x + 2)_*\right) + \left(x \cdot x + x\right))_*}}{\left(x - 1\right) \cdot (x * x + x)_*} \leadsto \frac{\color{blue}{\left(x - 1\right) \cdot \left(x - (2 * x + 2)_*\right) + \left(x \cdot x + x\right)}}{\left(x - 1\right) \cdot (x * x + x)_*}\]
      0.1
  1. Removed slow pow expressions

Original test:


(lambda ((x default))
  #:name "NMSE problem 3.3.3"
  (+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1)))
  #:target
  (/ 2 (* x (- (sqr x) 1))))