Average Error: 29.8 → 0.2
Time: 27.8s
Precision: 64
Internal Precision: 1408
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;(\left(1 + \frac{3}{x}\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_* \le -2.783723314922549 \cdot 10^{-10}:\\ \;\;\;\;\frac{x}{1 + x} - \frac{\sqrt{1 + x}}{1} \cdot \frac{\sqrt{1 + x}}{x - 1}\\ \mathbf{if}\;(\left(1 + \frac{3}{x}\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_* \le 1.0585824989560083 \cdot 10^{-11}:\\ \;\;\;\;(\left(1 + \frac{3}{x}\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{x}{x \cdot x - 1}\right) \cdot \left(x - 1\right) + \left(\frac{-\left(1 + x\right)}{x - 1}\right))_*\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if (fma (+ 1 (/ 3 x)) (/ (- 1) (* x x)) (/ (- 3) x)) < -2.783723314922549e-10

    1. Initial program 0.4

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

    3. Applied *-un-lft-identity0.4

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

    4. Applied add-sqr-sqrt0.5

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

    5. Applied times-frac0.5

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

    if -2.783723314922549e-10 < (fma (+ 1 (/ 3 x)) (/ (- 1) (* x x)) (/ (- 3) x)) < 1.0585824989560083e-11

    1. Initial program 60.1

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

    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 1.0585824989560083e-11 < (fma (+ 1 (/ 3 x)) (/ (- 1) (* x x)) (/ (- 3) x))

    1. Initial program 0.5

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

    3. Applied flip-+0.5

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

    4. Applied associate-/r/0.5

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

    5. Applied fma-neg0.5

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

  4. Applied simplify0.2

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;(\left(1 + \frac{3}{x}\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_* \le -2.783723314922549 \cdot 10^{-10}:\\ \;\;\;\;\frac{x}{1 + x} - \frac{\sqrt{1 + x}}{1} \cdot \frac{\sqrt{1 + x}}{x - 1}\\ \mathbf{if}\;(\left(1 + \frac{3}{x}\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_* \le 1.0585824989560083 \cdot 10^{-11}:\\ \;\;\;\;(\left(1 + \frac{3}{x}\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{x}{x \cdot x - 1}\right) \cdot \left(x - 1\right) + \left(\frac{-\left(1 + x\right)}{x - 1}\right))_*\\ \end{array}}\]

Runtime

Time bar (total: 27.8s)Debug logProfile

herbie shell --seed '#(1070100504 930361288 1279167582 284574201 1450237281 2578255382)' +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))