Average Error: 14.3 → 0.0
Time: 29.0s
Precision: 64
Internal Precision: 576
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{{x}^{3}} \le -1.399609321447887 \cdot 10^{-10}:\\ \;\;\;\;\frac{x}{(x \cdot x + 1)_*}\\ \mathbf{if}\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{{x}^{3}} \le 4.4266542066979285 \cdot 10^{-155}:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\sqrt{1^2 + x^2}^*}}{\sqrt{(x \cdot x + 1)_*}}\\ \end{array}\]

Error

Bits error versus x

Target

Original14.3
Target0.1
Herbie0.0
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Split input into 3 regimes

  2. if (- (+ (/ 1 (pow x 5)) (/ 1 x)) (/ 1 (pow x 3))) < -1.399609321447887e-10

    1. Initial program 0.1

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

    2. Applied simplify0.1

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

    if -1.399609321447887e-10 < (- (+ (/ 1 (pow x 5)) (/ 1 x)) (/ 1 (pow x 3))) < 4.4266542066979285e-155

    1. Initial program 39.6

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

    2. Applied simplify39.6

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

    3. Taylor expanded around inf 0

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

    if 4.4266542066979285e-155 < (- (+ (/ 1 (pow x 5)) (/ 1 x)) (/ 1 (pow x 3)))

    1. Initial program 0.1

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

    2. Applied simplify0.1

      \[\leadsto \color{blue}{\frac{x}{(x \cdot x + 1)_*}}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt0.1

      \[\leadsto \frac{x}{\color{blue}{\sqrt{(x \cdot x + 1)_*} \cdot \sqrt{(x \cdot x + 1)_*}}}\]

    5. Applied associate-/r*0.0

      \[\leadsto \color{blue}{\frac{\frac{x}{\sqrt{(x \cdot x + 1)_*}}}{\sqrt{(x \cdot x + 1)_*}}}\]

    6. Applied simplify0.0

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

Runtime

Time bar (total: 29.0s)Debug logProfile

herbie shell --seed 2018170 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

  :herbie-target
  (/ 1 (+ x (/ 1 x)))

  (/ x (+ (* x x) 1)))