\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
Test:
NMSE problem 3.3.3
Bits:
128 bits
Bits error versus x
Time: 10.8 s
Input Error: 4.2
Output Error: 0.2
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{(\left(\frac{1}{x} - 1\right) * \left(\left(\frac{1}{x} - \frac{2}{x}\right) - 2\right) + \left(\frac{1}{{x}^2} + \frac{1}{x}\right))_*}{\left(x - 1\right) \cdot (x * x + x)_*} & \text{when } x \le -0.0019466438f0 \\ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} & \text{when } x \le 0.52187485f0 \\ \frac{(\left(\frac{1}{x} - 1\right) * \left(\left(\frac{1}{x} - \frac{2}{x}\right) - 2\right) + \left(\frac{1}{{x}^2} + \frac{1}{x}\right))_*}{\left(x - 1\right) \cdot (x * x + x)_*} & \text{otherwise} \end{cases}\)

    if x < -0.0019466438f0

    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
      \[\color{red}{\left(\frac{1}{x + 1} - \frac{2}{x}\right)} + \frac{1}{x - 1} \leadsto \color{blue}{\frac{1 \cdot x - \left(x + 1\right) \cdot 2}{\left(x + 1\right) \cdot x}} + \frac{1}{x - 1}\]
      21.7
    4. Applied frac-add to get
      \[\color{red}{\frac{1 \cdot x - \left(x + 1\right) \cdot 2}{\left(x + 1\right) \cdot x} + \frac{1}{x - 1}} \leadsto \color{blue}{\frac{\left(1 \cdot x - \left(x + 1\right) \cdot 2\right) \cdot \left(x - 1\right) + \left(\left(x + 1\right) \cdot x\right) \cdot 1}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}}\]
      21.3
    5. Applied simplify to get
      \[\frac{\color{red}{\left(1 \cdot x - \left(x + 1\right) \cdot 2\right) \cdot \left(x - 1\right) + \left(\left(x + 1\right) \cdot x\right) \cdot 1}}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)} \leadsto \frac{\color{blue}{(\left(x - 1\right) * \left(x - (2 * x + 2)_*\right) + \left(x \cdot x + x\right))_*}}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}\]
      22.2
    6. Applied simplify to get
      \[\frac{(\left(x - 1\right) * \left(x - (2 * x + 2)_*\right) + \left(x \cdot x + x\right))_*}{\color{red}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}} \leadsto \frac{(\left(x - 1\right) * \left(x - (2 * x + 2)_*\right) + \left(x \cdot x + x\right))_*}{\color{blue}{\left(x - 1\right) \cdot (x * x + x)_*}}\]
      26.0
    7. Applied taylor to get
      \[\frac{(\left(x - 1\right) * \left(x - (2 * x + 2)_*\right) + \left(x \cdot x + x\right))_*}{\left(x - 1\right) \cdot (x * x + x)_*} \leadsto \frac{(\left(\frac{1}{x} - 1\right) * \left(\frac{1}{x} - (2 * \left(\frac{1}{x}\right) + 2)_*\right) + \left(\frac{1}{{x}^2} + \frac{1}{x}\right))_*}{\left(x - 1\right) \cdot (x * x + x)_*}\]
      0.6
    8. Taylor expanded around inf to get
      \[\frac{\color{red}{(\left(\frac{1}{x} - 1\right) * \left(\frac{1}{x} - (2 * \left(\frac{1}{x}\right) + 2)_*\right) + \left(\frac{1}{{x}^2} + \frac{1}{x}\right))_*}}{\left(x - 1\right) \cdot (x * x + x)_*} \leadsto \frac{\color{blue}{(\left(\frac{1}{x} - 1\right) * \left(\frac{1}{x} - (2 * \left(\frac{1}{x}\right) + 2)_*\right) + \left(\frac{1}{{x}^2} + \frac{1}{x}\right))_*}}{\left(x - 1\right) \cdot (x * x + x)_*}\]
      0.6
    9. Using strategy rm
      0.6
    10. Applied fma-udef to get
      \[\frac{(\left(\frac{1}{x} - 1\right) * \left(\frac{1}{x} - \color{red}{(2 * \left(\frac{1}{x}\right) + 2)_*}\right) + \left(\frac{1}{{x}^2} + \frac{1}{x}\right))_*}{\left(x - 1\right) \cdot (x * x + x)_*} \leadsto \frac{(\left(\frac{1}{x} - 1\right) * \left(\frac{1}{x} - \color{blue}{\left(2 \cdot \frac{1}{x} + 2\right)}\right) + \left(\frac{1}{{x}^2} + \frac{1}{x}\right))_*}{\left(x - 1\right) \cdot (x * x + x)_*}\]
      0.6
    11. Applied associate--r+ to get
      \[\frac{(\left(\frac{1}{x} - 1\right) * \color{red}{\left(\frac{1}{x} - \left(2 \cdot \frac{1}{x} + 2\right)\right)} + \left(\frac{1}{{x}^2} + \frac{1}{x}\right))_*}{\left(x - 1\right) \cdot (x * x + x)_*} \leadsto \frac{(\left(\frac{1}{x} - 1\right) * \color{blue}{\left(\left(\frac{1}{x} - 2 \cdot \frac{1}{x}\right) - 2\right)} + \left(\frac{1}{{x}^2} + \frac{1}{x}\right))_*}{\left(x - 1\right) \cdot (x * x + x)_*}\]
      0.6
    12. Applied simplify to get
      \[\frac{(\left(\frac{1}{x} - 1\right) * \left(\color{red}{\left(\frac{1}{x} - 2 \cdot \frac{1}{x}\right)} - 2\right) + \left(\frac{1}{{x}^2} + \frac{1}{x}\right))_*}{\left(x - 1\right) \cdot (x * x + x)_*} \leadsto \frac{(\left(\frac{1}{x} - 1\right) * \left(\color{blue}{\left(\frac{1}{x} - \frac{2}{x}\right)} - 2\right) + \left(\frac{1}{{x}^2} + \frac{1}{x}\right))_*}{\left(x - 1\right) \cdot (x * x + x)_*}\]
      0.6

    if -0.0019466438f0 < x < 0.52187485f0

    1. Started with
      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
      0.0

    if 0.52187485f0 < x

    1. Started with
      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
      8.0
    2. Using strategy rm
      8.0
    3. Applied frac-sub to get
      \[\color{red}{\left(\frac{1}{x + 1} - \frac{2}{x}\right)} + \frac{1}{x - 1} \leadsto \color{blue}{\frac{1 \cdot x - \left(x + 1\right) \cdot 2}{\left(x + 1\right) \cdot x}} + \frac{1}{x - 1}\]
      23.0
    4. Applied frac-add to get
      \[\color{red}{\frac{1 \cdot x - \left(x + 1\right) \cdot 2}{\left(x + 1\right) \cdot x} + \frac{1}{x - 1}} \leadsto \color{blue}{\frac{\left(1 \cdot x - \left(x + 1\right) \cdot 2\right) \cdot \left(x - 1\right) + \left(\left(x + 1\right) \cdot x\right) \cdot 1}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}}\]
      22.7
    5. Applied simplify to get
      \[\frac{\color{red}{\left(1 \cdot x - \left(x + 1\right) \cdot 2\right) \cdot \left(x - 1\right) + \left(\left(x + 1\right) \cdot x\right) \cdot 1}}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)} \leadsto \frac{\color{blue}{(\left(x - 1\right) * \left(x - (2 * x + 2)_*\right) + \left(x \cdot x + x\right))_*}}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}\]
      23.6
    6. Applied simplify to get
      \[\frac{(\left(x - 1\right) * \left(x - (2 * x + 2)_*\right) + \left(x \cdot x + x\right))_*}{\color{red}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}} \leadsto \frac{(\left(x - 1\right) * \left(x - (2 * x + 2)_*\right) + \left(x \cdot x + x\right))_*}{\color{blue}{\left(x - 1\right) \cdot (x * x + x)_*}}\]
      27.8
    7. Applied taylor to get
      \[\frac{(\left(x - 1\right) * \left(x - (2 * x + 2)_*\right) + \left(x \cdot x + x\right))_*}{\left(x - 1\right) \cdot (x * x + x)_*} \leadsto \frac{(\left(\frac{1}{x} - 1\right) * \left(\frac{1}{x} - (2 * \left(\frac{1}{x}\right) + 2)_*\right) + \left(\frac{1}{{x}^2} + \frac{1}{x}\right))_*}{\left(x - 1\right) \cdot (x * x + x)_*}\]
      0.2
    8. Taylor expanded around inf to get
      \[\frac{\color{red}{(\left(\frac{1}{x} - 1\right) * \left(\frac{1}{x} - (2 * \left(\frac{1}{x}\right) + 2)_*\right) + \left(\frac{1}{{x}^2} + \frac{1}{x}\right))_*}}{\left(x - 1\right) \cdot (x * x + x)_*} \leadsto \frac{\color{blue}{(\left(\frac{1}{x} - 1\right) * \left(\frac{1}{x} - (2 * \left(\frac{1}{x}\right) + 2)_*\right) + \left(\frac{1}{{x}^2} + \frac{1}{x}\right))_*}}{\left(x - 1\right) \cdot (x * x + x)_*}\]
      0.2
    9. Using strategy rm
      0.2
    10. Applied fma-udef to get
      \[\frac{(\left(\frac{1}{x} - 1\right) * \left(\frac{1}{x} - \color{red}{(2 * \left(\frac{1}{x}\right) + 2)_*}\right) + \left(\frac{1}{{x}^2} + \frac{1}{x}\right))_*}{\left(x - 1\right) \cdot (x * x + x)_*} \leadsto \frac{(\left(\frac{1}{x} - 1\right) * \left(\frac{1}{x} - \color{blue}{\left(2 \cdot \frac{1}{x} + 2\right)}\right) + \left(\frac{1}{{x}^2} + \frac{1}{x}\right))_*}{\left(x - 1\right) \cdot (x * x + x)_*}\]
      0.2
    11. Applied associate--r+ to get
      \[\frac{(\left(\frac{1}{x} - 1\right) * \color{red}{\left(\frac{1}{x} - \left(2 \cdot \frac{1}{x} + 2\right)\right)} + \left(\frac{1}{{x}^2} + \frac{1}{x}\right))_*}{\left(x - 1\right) \cdot (x * x + x)_*} \leadsto \frac{(\left(\frac{1}{x} - 1\right) * \color{blue}{\left(\left(\frac{1}{x} - 2 \cdot \frac{1}{x}\right) - 2\right)} + \left(\frac{1}{{x}^2} + \frac{1}{x}\right))_*}{\left(x - 1\right) \cdot (x * x + x)_*}\]
      0.1
    12. Applied simplify to get
      \[\frac{(\left(\frac{1}{x} - 1\right) * \left(\color{red}{\left(\frac{1}{x} - 2 \cdot \frac{1}{x}\right)} - 2\right) + \left(\frac{1}{{x}^2} + \frac{1}{x}\right))_*}{\left(x - 1\right) \cdot (x * x + x)_*} \leadsto \frac{(\left(\frac{1}{x} - 1\right) * \left(\color{blue}{\left(\frac{1}{x} - \frac{2}{x}\right)} - 2\right) + \left(\frac{1}{{x}^2} + \frac{1}{x}\right))_*}{\left(x - 1\right) \cdot (x * x + x)_*}\]
      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))))