Average Error: 29.3 → 0.9
Time: 1.8m
Precision: 64
Internal Precision: 1344
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;(\left(1 + \frac{3}{x}\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_* \le -1.1355472292343013 \cdot 10^{-23}:\\ \;\;\;\;\frac{x}{1 + x} - \frac{(\left(1 + x\right) \cdot \left((x \cdot x + x)_*\right) + \left(1 + x\right))_*}{(\left(x \cdot x\right) \cdot x + \left(-1\right))_*}\\ \mathbf{if}\;(\left(1 + \frac{3}{x}\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_* \le 1.2157089492426344 \cdot 10^{-08}:\\ \;\;\;\;(\left(1 + \frac{3}{x}\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;(x \cdot \left(\frac{1}{1 + x}\right) + \left(-\frac{1 + x}{x - 1}\right))_*\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if (fma (+ 1 (/ 3 x)) (/ (- 1) (* x x)) (- (/ 3 x))) < -1.1355472292343013e-23

    1. Initial program 3.0

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

    3. Applied flip3--3.0

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

    4. Applied associate-/r/3.0

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

    5. Applied add-cube-cbrt3.1

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

    6. Applied prod-diff3.0

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

    7. Applied simplify2.9

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

    8. Applied simplify3.0

      \[\leadsto \left(\frac{x}{1 + x} - \frac{(\left(1 + x\right) \cdot \left((x \cdot x + x)_*\right) + \left(1 + x\right))_*}{(\left(x \cdot x\right) \cdot x + \left(-1\right))_*}\right) + \color{blue}{0}\]

    if -1.1355472292343013e-23 < (fma (+ 1 (/ 3 x)) (/ (- 1) (* x x)) (- (/ 3 x))) < 1.2157089492426344e-08

    1. Initial program 60.2

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

    2. Taylor expanded around inf 0.3

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

    3. Applied simplify0.0

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

    if 1.2157089492426344e-08 < (fma (+ 1 (/ 3 x)) (/ (- 1) (* x x)) (- (/ 3 x)))

    1. Initial program 0.3

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

    3. Applied div-inv0.3

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

    4. Applied fma-neg0.3

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

  4. Applied simplify0.9

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;(\left(1 + \frac{3}{x}\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_* \le -1.1355472292343013 \cdot 10^{-23}:\\ \;\;\;\;\frac{x}{1 + x} - \frac{(\left(1 + x\right) \cdot \left((x \cdot x + x)_*\right) + \left(1 + x\right))_*}{(\left(x \cdot x\right) \cdot x + \left(-1\right))_*}\\ \mathbf{if}\;(\left(1 + \frac{3}{x}\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_* \le 1.2157089492426344 \cdot 10^{-08}:\\ \;\;\;\;(\left(1 + \frac{3}{x}\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;(x \cdot \left(\frac{1}{1 + x}\right) + \left(-\frac{1 + x}{x - 1}\right))_*\\ \end{array}}\]

Runtime

Time bar (total: 1.8m)Debug logProfile

herbie shell --seed 2018193 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))