\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
Test:
NMSE problem 3.3.3
Bits:
128 bits
Bits error versus x
Time: 27.9 s
Input Error: 4.1
Output Error: 0.1
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{(\left(\frac{1}{x} - 1\right) * \left(\frac{1}{x} - (2 * \left(\frac{1}{x}\right) + 2)_*\right) + \left(\frac{1}{x} + \frac{\frac{1}{x}}{x}\right))_*}{\left(x - 1\right) \cdot (x * x + x)_*} & \text{when } x \le -2.3631973f0 \\ \left(\left(\frac{1}{x - 1} + \frac{1}{1 + x}\right) - \frac{2}{x}\right) \cdot 1 & \text{when } x \le 3.1920562f0 \\ \frac{(\left(\frac{1}{x} - 1\right) * \left(\frac{1}{x} - (2 * \left(\frac{1}{x}\right) + 2)_*\right) + \left(\frac{1}{x} + \frac{\frac{1}{x}}{x}\right))_*}{\left(x - 1\right) \cdot (x * x + x)_*} & \text{otherwise} \end{cases}\)

    if x < -2.3631973f0

    1. Started with
      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
      8.5
    2. Using strategy rm
      8.5
    3. Applied frac-sub to get
      \[\color{red}{\left(\frac{1}{x + 1} - \frac{2}{x}\right)} + \frac{1}{x - 1} \leadsto \color{blue}{\frac{1 \cdot x - \left(x + 1\right) \cdot 2}{\left(x + 1\right) \cdot x}} + \frac{1}{x - 1}\]
      23.5
    4. Applied frac-add to get
      \[\color{red}{\frac{1 \cdot x - \left(x + 1\right) \cdot 2}{\left(x + 1\right) \cdot x} + \frac{1}{x - 1}} \leadsto \color{blue}{\frac{\left(1 \cdot x - \left(x + 1\right) \cdot 2\right) \cdot \left(x - 1\right) + \left(\left(x + 1\right) \cdot x\right) \cdot 1}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}}\]
      23.2
    5. Applied simplify to get
      \[\frac{\color{red}{\left(1 \cdot x - \left(x + 1\right) \cdot 2\right) \cdot \left(x - 1\right) + \left(\left(x + 1\right) \cdot x\right) \cdot 1}}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)} \leadsto \frac{\color{blue}{(\left(x - 1\right) * \left(x - (2 * x + 2)_*\right) + \left(x \cdot x + x\right))_*}}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}\]
      24.2
    6. Applied simplify to get
      \[\frac{(\left(x - 1\right) * \left(x - (2 * x + 2)_*\right) + \left(x \cdot x + x\right))_*}{\color{red}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}} \leadsto \frac{(\left(x - 1\right) * \left(x - (2 * x + 2)_*\right) + \left(x \cdot x + x\right))_*}{\color{blue}{\left(x - 1\right) \cdot (x * x + x)_*}}\]
      28.2
    7. Applied taylor to get
      \[\frac{(\left(x - 1\right) * \left(x - (2 * x + 2)_*\right) + \left(x \cdot x + x\right))_*}{\left(x - 1\right) \cdot (x * x + x)_*} \leadsto \frac{(\left(\frac{1}{x} - 1\right) * \left(\frac{1}{x} - (2 * \left(\frac{1}{x}\right) + 2)_*\right) + \left(\frac{1}{{x}^2} + \frac{1}{x}\right))_*}{\left(x - 1\right) \cdot (x * x + x)_*}\]
      0.1
    8. Taylor expanded around inf to get
      \[\frac{\color{red}{(\left(\frac{1}{x} - 1\right) * \left(\frac{1}{x} - (2 * \left(\frac{1}{x}\right) + 2)_*\right) + \left(\frac{1}{{x}^2} + \frac{1}{x}\right))_*}}{\left(x - 1\right) \cdot (x * x + x)_*} \leadsto \frac{\color{blue}{(\left(\frac{1}{x} - 1\right) * \left(\frac{1}{x} - (2 * \left(\frac{1}{x}\right) + 2)_*\right) + \left(\frac{1}{{x}^2} + \frac{1}{x}\right))_*}}{\left(x - 1\right) \cdot (x * x + x)_*}\]
      0.1
    9. Applied simplify to get
      \[\frac{(\left(\frac{1}{x} - 1\right) * \left(\frac{1}{x} - (2 * \left(\frac{1}{x}\right) + 2)_*\right) + \left(\frac{1}{{x}^2} + \frac{1}{x}\right))_*}{\left(x - 1\right) \cdot (x * x + x)_*} \leadsto \frac{(\left(\frac{1}{x} - 1\right) * \left(\frac{1}{x} - (2 * \left(\frac{1}{x}\right) + 2)_*\right) + \left(\frac{1}{x} + \frac{\frac{1}{x}}{x}\right))_*}{\left(x - 1\right) \cdot (x * x + x)_*}\]
      0.1
    10. Applied final simplification

    if -2.3631973f0 < x < 3.1920562f0

    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 flip-+ to get
      \[\color{red}{\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}} \leadsto \color{blue}{\frac{{\left(\frac{1}{x + 1} - \frac{2}{x}\right)}^2 - {\left(\frac{1}{x - 1}\right)}^2}{\left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{1}{x - 1}}}\]
      13.9
    4. Using strategy rm
      13.9
    5. Applied *-un-lft-identity to get
      \[\frac{{\left(\frac{1}{x + 1} - \frac{2}{x}\right)}^2 - {\left(\frac{1}{x - 1}\right)}^2}{\color{red}{\left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{1}{x - 1}}} \leadsto \frac{{\left(\frac{1}{x + 1} - \frac{2}{x}\right)}^2 - {\left(\frac{1}{x - 1}\right)}^2}{\color{blue}{1 \cdot \left(\left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{1}{x - 1}\right)}}\]
      13.9
    6. Applied difference-of-squares to get
      \[\frac{\color{red}{{\left(\frac{1}{x + 1} - \frac{2}{x}\right)}^2 - {\left(\frac{1}{x - 1}\right)}^2}}{1 \cdot \left(\left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{1}{x - 1}\right)} \leadsto \frac{\color{blue}{\left(\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\right) \cdot \left(\left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{1}{x - 1}\right)}}{1 \cdot \left(\left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{1}{x - 1}\right)}\]
      13.9
    7. Applied times-frac to get
      \[\color{red}{\frac{\left(\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\right) \cdot \left(\left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{1}{x - 1}\right)}{1 \cdot \left(\left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{1}{x - 1}\right)}} \leadsto \color{blue}{\frac{\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}}{1} \cdot \frac{\left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{1}{x - 1}}{\left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{1}{x - 1}}}\]
      0.1
    8. Applied simplify to get
      \[\color{red}{\frac{\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}}{1}} \cdot \frac{\left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{1}{x - 1}}{\left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{1}{x - 1}} \leadsto \color{blue}{\left(\left(\frac{1}{x - 1} + \frac{1}{1 + x}\right) - \frac{2}{x}\right)} \cdot \frac{\left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{1}{x - 1}}{\left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{1}{x - 1}}\]
      0.0
    9. Applied simplify to get
      \[\left(\left(\frac{1}{x - 1} + \frac{1}{1 + x}\right) - \frac{2}{x}\right) \cdot \color{red}{\frac{\left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{1}{x - 1}}{\left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{1}{x - 1}}} \leadsto \left(\left(\frac{1}{x - 1} + \frac{1}{1 + x}\right) - \frac{2}{x}\right) \cdot \color{blue}{1}\]
      0.0

    if 3.1920562f0 < x

    1. Started with
      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
      8.1
    2. Using strategy rm
      8.1
    3. Applied frac-sub to get
      \[\color{red}{\left(\frac{1}{x + 1} - \frac{2}{x}\right)} + \frac{1}{x - 1} \leadsto \color{blue}{\frac{1 \cdot x - \left(x + 1\right) \cdot 2}{\left(x + 1\right) \cdot x}} + \frac{1}{x - 1}\]
      23.5
    4. Applied frac-add to get
      \[\color{red}{\frac{1 \cdot x - \left(x + 1\right) \cdot 2}{\left(x + 1\right) \cdot x} + \frac{1}{x - 1}} \leadsto \color{blue}{\frac{\left(1 \cdot x - \left(x + 1\right) \cdot 2\right) \cdot \left(x - 1\right) + \left(\left(x + 1\right) \cdot x\right) \cdot 1}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}}\]
      23.3
    5. Applied simplify to get
      \[\frac{\color{red}{\left(1 \cdot x - \left(x + 1\right) \cdot 2\right) \cdot \left(x - 1\right) + \left(\left(x + 1\right) \cdot x\right) \cdot 1}}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)} \leadsto \frac{\color{blue}{(\left(x - 1\right) * \left(x - (2 * x + 2)_*\right) + \left(x \cdot x + x\right))_*}}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}\]
      24.2
    6. Applied simplify to get
      \[\frac{(\left(x - 1\right) * \left(x - (2 * x + 2)_*\right) + \left(x \cdot x + x\right))_*}{\color{red}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}} \leadsto \frac{(\left(x - 1\right) * \left(x - (2 * x + 2)_*\right) + \left(x \cdot x + x\right))_*}{\color{blue}{\left(x - 1\right) \cdot (x * x + x)_*}}\]
      28.5
    7. Applied taylor to get
      \[\frac{(\left(x - 1\right) * \left(x - (2 * x + 2)_*\right) + \left(x \cdot x + x\right))_*}{\left(x - 1\right) \cdot (x * x + x)_*} \leadsto \frac{(\left(\frac{1}{x} - 1\right) * \left(\frac{1}{x} - (2 * \left(\frac{1}{x}\right) + 2)_*\right) + \left(\frac{1}{{x}^2} + \frac{1}{x}\right))_*}{\left(x - 1\right) \cdot (x * x + x)_*}\]
      0.1
    8. Taylor expanded around inf to get
      \[\frac{\color{red}{(\left(\frac{1}{x} - 1\right) * \left(\frac{1}{x} - (2 * \left(\frac{1}{x}\right) + 2)_*\right) + \left(\frac{1}{{x}^2} + \frac{1}{x}\right))_*}}{\left(x - 1\right) \cdot (x * x + x)_*} \leadsto \frac{\color{blue}{(\left(\frac{1}{x} - 1\right) * \left(\frac{1}{x} - (2 * \left(\frac{1}{x}\right) + 2)_*\right) + \left(\frac{1}{{x}^2} + \frac{1}{x}\right))_*}}{\left(x - 1\right) \cdot (x * x + x)_*}\]
      0.1
    9. Applied simplify to get
      \[\frac{(\left(\frac{1}{x} - 1\right) * \left(\frac{1}{x} - (2 * \left(\frac{1}{x}\right) + 2)_*\right) + \left(\frac{1}{{x}^2} + \frac{1}{x}\right))_*}{\left(x - 1\right) \cdot (x * x + x)_*} \leadsto \frac{(\left(\frac{1}{x} - 1\right) * \left(\frac{1}{x} - (2 * \left(\frac{1}{x}\right) + 2)_*\right) + \left(\frac{1}{x} + \frac{\frac{1}{x}}{x}\right))_*}{\left(x - 1\right) \cdot (x * x + x)_*}\]
      0.1
    10. Applied final simplification
  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))))