Average Error: 29.5 → 0.1
Time: 5.3m
Precision: 64
Internal Precision: 1344
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;(\left(1 + \frac{3}{x}\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_* \le -1.2826091876195137 \cdot 10^{-06}:\\ \;\;\;\;(e^{\log_* (1 + \frac{x}{1 + x})} - 1)^* - \frac{1 + x}{x - 1}\\ \mathbf{if}\;(\left(1 + \frac{3}{x}\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_* \le 9.763969376861291 \cdot 10^{-08}:\\ \;\;\;\;(\left(1 + \frac{3}{x}\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x} - \frac{(\left(1 + x\right) \cdot \left((x \cdot x + x)_*\right) + \left(1 + x\right))_*}{(\left(x \cdot x\right) \cdot x + \left(-1\right))_*}\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if (fma (+ 1 (/ 3 x)) (/ (- 1) (* x x)) (- (/ 3 x))) < -1.2826091876195137e-06

    1. Initial program 0.1

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

    3. Applied expm1-log1p-u0.1

      \[\leadsto \color{blue}{(e^{\log_* (1 + \frac{x}{x + 1})} - 1)^*} - \frac{x + 1}{x - 1}\]

    if -1.2826091876195137e-06 < (fma (+ 1 (/ 3 x)) (/ (- 1) (* x x)) (- (/ 3 x))) < 9.763969376861291e-08

    1. Initial program 59.7

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

    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 9.763969376861291e-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 flip3--0.3

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

    4. Applied associate-/r/0.3

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

    5. Applied add-cube-cbrt0.3

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

    6. Applied prod-diff0.3

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

    7. Applied simplify0.3

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

    8. Applied simplify0.3

      \[\leadsto \left(\frac{x}{1 + x} - \frac{(\left(1 + x\right) \cdot \left((x \cdot x + x)_*\right) + \left(1 + x\right))_*}{(\left(x \cdot x\right) \cdot x + \left(-1\right))_*}\right) + \color{blue}{0}\]
  3. Recombined 3 regimes into one program.

  4. Applied simplify0.1

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;(\left(1 + \frac{3}{x}\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_* \le -1.2826091876195137 \cdot 10^{-06}:\\ \;\;\;\;(e^{\log_* (1 + \frac{x}{1 + x})} - 1)^* - \frac{1 + x}{x - 1}\\ \mathbf{if}\;(\left(1 + \frac{3}{x}\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_* \le 9.763969376861291 \cdot 10^{-08}:\\ \;\;\;\;(\left(1 + \frac{3}{x}\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x} - \frac{(\left(1 + x\right) \cdot \left((x \cdot x + x)_*\right) + \left(1 + x\right))_*}{(\left(x \cdot x\right) \cdot x + \left(-1\right))_*}\\ \end{array}}\]

Runtime

Time bar (total: 5.3m)Debug logProfile

herbie shell --seed '#(1071821486 549052472 3784827256 1559736200 3548510075 881134285)' +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))