Average Error: 14.6 → 0.1
Time: 11.2s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\frac{1}{\frac{1}{x} + x}\]
\frac{x}{x \cdot x + 1}
↓
\frac{1}{\frac{1}{x} + x}
double f(double x) {
        double r47031 = x;
        double r47032 = r47031 * r47031;
        double r47033 = 1.0;
        double r47034 = r47032 + r47033;
        double r47035 = r47031 / r47034;
        return r47035;
}
↓
double f(double x) {
        double r47036 = 1.0;
        double r47037 = 1.0;
        double r47038 = x;
        double r47039 = r47037 / r47038;
        double r47040 = r47039 + r47038;
        double r47041 = r47036 / r47040;
        return r47041;
}

Error

Bits error versus x

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Results

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Target

Original14.6
Target0.1
Herbie0.1
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Initial program 14.6

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

  3. Applied add-sqr-sqrt14.6

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

  4. Applied associate-/r*14.6

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

  5. Simplified14.6

    \[\leadsto \frac{\color{blue}{\frac{x}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}}}{\sqrt{x \cdot x + 1}}\]
  6. Using strategy rm

  7. Applied *-un-lft-identity14.6

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

  8. Applied sqrt-prod14.6

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

  9. Applied *-un-lft-identity14.6

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

  10. Applied times-frac14.6

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

  11. Applied associate-/l*14.7

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

  12. Simplified14.7

    \[\leadsto \frac{\frac{1}{\sqrt{1}}}{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}{x} \cdot \sqrt{\mathsf{fma}\left(x, x, 1\right)}}}\]

  13. Taylor expanded around 0 0.1

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

  14. Simplified0.1

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

  15. Final simplification0.1

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

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

  :herbie-target
  (/ 1.0 (+ x (/ 1.0 x)))

  (/ x (+ (* x x) 1.0)))