Average Error: 10.9 → 0.1
Time: 47.6s
Precision: 64
Internal Precision: 1152
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.7387217572195604 \cdot 10^{-16}:\\ \;\;\;\;\left(\frac{2}{{x}^{7}} + \frac{2}{{x}^{5}}\right) + \frac{\frac{\frac{2}{x}}{x}}{x}\\ \mathbf{if}\;x \le 2500116.352981572:\\ \;\;\;\;-\left(2 \cdot \frac{1}{x} + \left(2 \cdot {x}^{3} + 2 \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{{x}^{7}} + \frac{2}{{x}^{5}}\right) + \frac{\frac{2}{x}}{x \cdot x}\\ \end{array}\]

Error

Bits error versus x

Target

Original10.9
Target1.6
Herbie0.1
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.7387217572195604e-16

    1. Initial program 21.9

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

    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. Applied simplify0.1

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

    5. Applied associate-/r*0.1

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

    if -1.7387217572195604e-16 < x < 2500116.352981572

    1. Initial program 0.7

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

    2. Taylor expanded around 0 0.0

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

    if 2500116.352981572 < x

    1. Initial program 19.2

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

    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. Applied simplify0.1

      \[\leadsto \color{blue}{\left(\frac{2}{{x}^{7}} + \frac{2}{{x}^{5}}\right) + \frac{\frac{2}{x}}{x \cdot x}}\]
  3. Recombined 3 regimes into one program.
  4. Removed slow pow expressions.

Runtime

Time bar (total: 47.6s)Debug logProfile

herbie shell --seed '#(1062803647 245428163 493620569 3595423923 1908391097 2390014376)' 
(FPCore (x)
  :name "3frac (problem 3.3.3)"

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

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