Error

Target

Derivation

1. Split input into 2 regimes

2. if (+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1))) < -3.8138284005463116e-18 or 1.3250891055449946e-27 < (+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1)))

   1. Initial program 0.7

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

   2. Initial simplification0.7

      \[\leadsto \left(\frac{1}{x - 1} + \frac{1}{x + 1}\right) - \frac{2}{x}\]
   3. Using strategy rm

   4. Applied frac-add0.7

      \[\leadsto \color{blue}{\frac{1 \cdot \left(x + 1\right) + \left(x - 1\right) \cdot 1}{\left(x - 1\right) \cdot \left(x + 1\right)}} - \frac{2}{x}\]

   5. Applied frac-sub0.2

      \[\leadsto \color{blue}{\frac{\left(1 \cdot \left(x + 1\right) + \left(x - 1\right) \cdot 1\right) \cdot x - \left(\left(x - 1\right) \cdot \left(x + 1\right)\right) \cdot 2}{\left(\left(x - 1\right) \cdot \left(x + 1\right)\right) \cdot x}}\]

   6. Simplified0.2

      \[\leadsto \frac{\color{blue}{(\left((x \cdot \left(x - 1\right) + \left(x - 1\right))_*\right) \cdot \left(-2\right) + \left((x \cdot \left(x - 0\right) + \left(x \cdot x\right))_*\right))_*}}{\left(\left(x - 1\right) \cdot \left(x + 1\right)\right) \cdot x}\]

   7. Simplified0.2

      \[\leadsto \frac{(\left((x \cdot \left(x - 1\right) + \left(x - 1\right))_*\right) \cdot \left(-2\right) + \left((x \cdot \left(x - 0\right) + \left(x \cdot x\right))_*\right))_*}{\color{blue}{(x \cdot x + x)_* \cdot \left(x - 1\right)}}\]

   if -3.8138284005463116e-18 < (+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1))) < 1.3250891055449946e-27

   1. Initial program 19.3

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

   2. Taylor expanded around inf 0.5

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{x}^{3}} + \left(2 \cdot \frac{1}{{x}^{5}} + 2 \cdot \frac{1}{{x}^{7}}\right)}\]

   3. Simplified0.1

      \[\leadsto \color{blue}{\left(\frac{2}{{x}^{7}} + \frac{2}{{x}^{5}}\right) + \frac{\frac{2}{x}}{x \cdot x}}\]
   4. Using strategy rm

   5. Applied add-log-exp0.1

      \[\leadsto \left(\frac{2}{{x}^{7}} + \color{blue}{\log \left(e^{\frac{2}{{x}^{5}}}\right)}\right) + \frac{\frac{2}{x}}{x \cdot x}\]
3. Recombined 2 regimes into one program.

4. Final simplification0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{x - 1} + \left(\frac{1}{x + 1} - \frac{2}{x}\right) \le -3.8138284005463116 \cdot 10^{-18}:\\ \;\;\;\;\frac{(\left((x \cdot \left(x - 1\right) + \left(x - 1\right))_*\right) \cdot \left(-2\right) + \left((x \cdot x + \left(x \cdot x\right))_*\right))_*}{\left(x - 1\right) \cdot (x \cdot x + x)_*}\\ \mathbf{elif}\;\frac{1}{x - 1} + \left(\frac{1}{x + 1} - \frac{2}{x}\right) \le 1.3250891055449946 \cdot 10^{-27}:\\ \;\;\;\;\left(\frac{2}{{x}^{7}} + \log \left(e^{\frac{2}{{x}^{5}}}\right)\right) + \frac{\frac{2}{x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left((x \cdot \left(x - 1\right) + \left(x - 1\right))_*\right) \cdot \left(-2\right) + \left((x \cdot x + \left(x \cdot x\right))_*\right))_*}{\left(x - 1\right) \cdot (x \cdot x + x)_*}\\ \end{array}\]

Runtime

Time bar (total: 43.2s)Debug logProfile

herbie shell --seed 2018215 +o rules:numerics
(FPCore (x)
  :name "3frac (problem 3.3.3)"

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

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