Average Error: 10.1 → 0.1
Time: 6.0m
Precision: 64
Internal Precision: 1088
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{x - 1} + \left(\frac{1}{x + 1} - \frac{2}{x}\right) \le -1.630811661951327 \cdot 10^{-05}:\\ \;\;\;\;\frac{(x \cdot \left(\left(\frac{1}{x - 1} + \frac{1}{x + 1}\right) \cdot \left(\frac{1}{x - 1} - \frac{1}{x + 1}\right)\right) + \left(\left(-2\right) \cdot \left(\frac{1}{x - 1} - \frac{1}{x + 1}\right)\right))_*}{\left(\frac{1}{x - 1} - \frac{1}{x + 1}\right) \cdot x}\\ \mathbf{elif}\;\frac{1}{x - 1} + \left(\frac{1}{x + 1} - \frac{2}{x}\right) \le 4.0031040743915335 \cdot 10^{-11}:\\ \;\;\;\;\left(\frac{2}{{x}^{5}} + \frac{2}{{x}^{7}}\right) + \frac{\frac{\frac{2}{x}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x + 1} + \frac{\frac{1}{\sqrt[3]{x - 1} \cdot \sqrt[3]{x - 1}}}{\sqrt[3]{x - 1}}\right) - \frac{2}{x}\\ \end{array}\]

Error

Bits error versus x

Target

Original10.1
Target0.3
Herbie0.1
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if (+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1))) < -1.630811661951327e-05

    1. Initial program 0.0

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

    2. Initial simplification0.0

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

    4. Applied add-cube-cbrt0.0

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

    5. Applied associate-/r*0.0

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

    7. Applied flip-+0.0

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

    8. Applied frac-sub0.0

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

    9. Simplified0.1

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

    10. Simplified0.0

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

    if -1.630811661951327e-05 < (+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1))) < 4.0031040743915335e-11

    1. Initial program 20.1

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

    2. Initial simplification20.1

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

    3. Taylor expanded around inf 0.6

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

    4. Simplified0.1

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

    6. Applied associate-/r*0.2

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

    if 4.0031040743915335e-11 < (+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1)))

    1. Initial program 0.1

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

    2. Initial simplification0.1

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

    4. Applied add-cube-cbrt0.1

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

    5. Applied associate-/r*0.1

      \[\leadsto \left(\color{blue}{\frac{\frac{1}{\sqrt[3]{x - 1} \cdot \sqrt[3]{x - 1}}}{\sqrt[3]{x - 1}}} + \frac{1}{x + 1}\right) - \frac{2}{x}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{x - 1} + \left(\frac{1}{x + 1} - \frac{2}{x}\right) \le -1.630811661951327 \cdot 10^{-05}:\\ \;\;\;\;\frac{(x \cdot \left(\left(\frac{1}{x - 1} + \frac{1}{x + 1}\right) \cdot \left(\frac{1}{x - 1} - \frac{1}{x + 1}\right)\right) + \left(\left(-2\right) \cdot \left(\frac{1}{x - 1} - \frac{1}{x + 1}\right)\right))_*}{\left(\frac{1}{x - 1} - \frac{1}{x + 1}\right) \cdot x}\\ \mathbf{elif}\;\frac{1}{x - 1} + \left(\frac{1}{x + 1} - \frac{2}{x}\right) \le 4.0031040743915335 \cdot 10^{-11}:\\ \;\;\;\;\left(\frac{2}{{x}^{5}} + \frac{2}{{x}^{7}}\right) + \frac{\frac{\frac{2}{x}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x + 1} + \frac{\frac{1}{\sqrt[3]{x - 1} \cdot \sqrt[3]{x - 1}}}{\sqrt[3]{x - 1}}\right) - \frac{2}{x}\\ \end{array}\]

Runtime

Time bar (total: 6.0m)Debug logProfile

herbie shell --seed 2018214 +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))))