Average Error: 28.8 → 0.4
Time: 3.8m
Precision: 64
Internal Precision: 1344
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;(\left(\frac{3}{x} + 1\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(-\frac{3}{x}\right))_* \le -9.35362369942047 \cdot 10^{-11}:\\ \;\;\;\;\frac{\left(\left(x - 1\right) - \left(1 + x\right)\right) - \frac{1 + x}{x}}{\left(x - 1\right) \cdot \frac{1 + x}{x}}\\ \mathbf{elif}\;(\left(\frac{3}{x} + 1\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(-\frac{3}{x}\right))_* \le 4.234656599064631 \cdot 10^{-25}:\\ \;\;\;\;(\left(\frac{3}{x} + 1\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(-\frac{3}{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(x - 1\right) - \left(1 + x\right)\right) - \frac{1 + x}{x}}{\frac{\left(1 + x\right) \cdot \left(x \cdot x - 1\right)}{(x \cdot x + x)_*}}\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if (fma (+ 1 (/ 3 x)) (/ (- 1) (* x x)) (/ (- 3) x)) < -9.35362369942047e-11

    1. Initial program 0.4

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
    2. Using strategy rm

    3. Applied clear-num0.4

      \[\leadsto \color{blue}{\frac{1}{\frac{x + 1}{x}}} - \frac{x + 1}{x - 1}\]
    4. Using strategy rm

    5. Applied frac-sub0.4

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

    6. Simplified0.0

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

    if -9.35362369942047e-11 < (fma (+ 1 (/ 3 x)) (/ (- 1) (* x x)) (/ (- 3) x)) < 4.234656599064631e-25

    1. Initial program 60.4

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]

    2. Taylor expanded around inf 0.3

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

    3. Simplified0.0

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

    if 4.234656599064631e-25 < (fma (+ 1 (/ 3 x)) (/ (- 1) (* x x)) (/ (- 3) x))

    1. Initial program 2.5

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
    2. Using strategy rm

    3. Applied clear-num2.5

      \[\leadsto \color{blue}{\frac{1}{\frac{x + 1}{x}}} - \frac{x + 1}{x - 1}\]
    4. Using strategy rm

    5. Applied frac-sub1.9

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

    6. Simplified1.2

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

    8. Applied flip--1.2

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

    9. Applied frac-times1.2

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

    10. Simplified1.2

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;(\left(\frac{3}{x} + 1\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(-\frac{3}{x}\right))_* \le -9.35362369942047 \cdot 10^{-11}:\\ \;\;\;\;\frac{\left(\left(x - 1\right) - \left(1 + x\right)\right) - \frac{1 + x}{x}}{\left(x - 1\right) \cdot \frac{1 + x}{x}}\\ \mathbf{elif}\;(\left(\frac{3}{x} + 1\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(-\frac{3}{x}\right))_* \le 4.234656599064631 \cdot 10^{-25}:\\ \;\;\;\;(\left(\frac{3}{x} + 1\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(-\frac{3}{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(x - 1\right) - \left(1 + x\right)\right) - \frac{1 + x}{x}}{\frac{\left(1 + x\right) \cdot \left(x \cdot x - 1\right)}{(x \cdot x + x)_*}}\\ \end{array}\]

Runtime

Time bar (total: 3.8m)Debug logProfile

herbie shell --seed 2018216 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))