Average Error: 15.1 → 0.0
Time: 33.7s
Precision: 64
Internal Precision: 576
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{{x}^{3}} \le -8.98475646888388 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{x}{\sqrt{1^2 + x^2}^*}}{\sqrt{x \cdot x + 1}}\\ \mathbf{if}\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{{x}^{3}} \le 3.9064990173073113 \cdot 10^{-25}:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{(x \cdot x + 1)_*}\\ \end{array}\]

Error

Bits error versus x

Target

Original15.1
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))) < -8.98475646888388e-17

    1. Initial program 0.1

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

    3. Applied add-sqr-sqrt0.1

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

    4. Applied associate-/r*0.1

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

    5. Applied simplify0.1

      \[\leadsto \frac{\color{blue}{\frac{x}{\sqrt{1^2 + x^2}^*}}}{\sqrt{x \cdot x + 1}}\]

    if -8.98475646888388e-17 < (- (+ (/ 1 (pow x 5)) (/ 1 x)) (/ 1 (pow x 3))) < 3.9064990173073113e-25

    1. Initial program 32.0

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

    2. Taylor expanded around inf 0

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

    if 3.9064990173073113e-25 < (- (+ (/ 1 (pow x 5)) (/ 1 x)) (/ 1 (pow x 3)))

    1. Initial program 0.0

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

    3. Applied div-inv0.0

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

    4. Applied simplify0.0

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

Runtime

Time bar (total: 33.7s)Debug logProfile

herbie shell --seed '#(1071979731 1496239409 439705970 2863295848 982327776 189749553)' +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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