Average Error: 9.7 → 0.2
Time: 23.3s
Precision: 64
Internal Precision: 1088
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{2}{{x}^{3} - x}\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.7
Target0.2
Herbie0.2
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Initial program 9.7

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

  3. Applied frac-sub26.4

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

  4. Applied frac-add25.7

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

  5. Simplified26.0

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

  6. Simplified26.0

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

  7. Taylor expanded around 0 0.2

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

  8. Taylor expanded around 0 0.2

    \[\leadsto \frac{2}{\color{blue}{{x}^{3} - x}}\]

  9. Final simplification0.2

    \[\leadsto \frac{2}{{x}^{3} - x}\]

Runtime

Time bar (total: 23.3s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes0.20.20.00.20%
herbie shell --seed 2018286 +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))))