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Target

Derivation

1. Split input into 3 regimes

2. if (- (+ (/ 1 (pow x 5)) (/ 1 x)) (/ 1 (pow x 3))) < -8.998151061271119e-07

   1. Initial program 0.0

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

   3. Applied clear-num0.2

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

   if -8.998151061271119e-07 < (- (+ (/ 1 (pow x 5)) (/ 1 x)) (/ 1 (pow x 3))) < 1.624868472513255e-158

   1. Initial program 40.6

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

   2. Taylor expanded around inf 0.0

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

   if 1.624868472513255e-158 < (- (+ (/ 1 (pow x 5)) (/ 1 x)) (/ 1 (pow x 3)))

   1. Initial program 0.4

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

   3. Applied add-sqr-sqrt0.4

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

   4. Applied associate-/r*0.4

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

   5. Simplified0.4

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

4. Final simplification0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - \frac{1}{{x}^{3}} \le -8.998151061271119 \cdot 10^{-07}:\\ \;\;\;\;\frac{1}{\frac{x \cdot x + 1}{x}}\\ \mathbf{elif}\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - \frac{1}{{x}^{3}} \le 1.624868472513255 \cdot 10^{-158}:\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\sqrt{x \cdot x + 1}}}{\sqrt{1^2 + x^2}^*}\\ \end{array}\]

Runtime

Time bar (total: 27.3s)Debug logProfile

herbie shell --seed 2018217 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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