Average Error: 9.9 → 0.8
Time: 1.5m
Precision: 64
Internal Precision: 1152
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \le -1990431009.4294772:\\ \;\;\;\;\left(\frac{\frac{1}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}}{\sqrt[3]{x + 1}} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \mathbf{if}\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \le 699.3919436253536:\\ \;\;\;\;\left(\frac{2}{{x}^{7}} + \frac{2}{{x}^{5}}\right) + \frac{\frac{2}{x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}}{\sqrt[3]{x + 1}} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if (+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1))) < -1990431009.4294772 or 699.3919436253536 < (+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1)))

    1. Initial program 0.0

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

    3. Applied add-cube-cbrt0.0

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

    4. Applied associate-/r*0.0

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

    if -1990431009.4294772 < (+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1))) < 699.3919436253536

    1. Initial program 19.3

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

    2. Taylor expanded around inf 2.0

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

    3. Applied simplify1.6

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

Runtime

Time bar (total: 1.5m)Debug log

herbie shell --seed '#(1743936871 1855164119 3668777427 1254258049 132811564 1366975197)' 
(FPCore (x)
  :name "NMSE problem 3.3.3"
  :pre (!= x 0 1 -1)
  (+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1))))