Average Error: 15.0 → 0.1
Time: 15.5s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\frac{1}{x + \frac{1}{x}}\]
\frac{x}{x \cdot x + 1}
↓
\frac{1}{x + \frac{1}{x}}
double f(double x) {
        double r1525318 = x;
        double r1525319 = r1525318 * r1525318;
        double r1525320 = 1.0;
        double r1525321 = r1525319 + r1525320;
        double r1525322 = r1525318 / r1525321;
        return r1525322;
}
↓
double f(double x) {
        double r1525323 = 1.0;
        double r1525324 = x;
        double r1525325 = r1525323 / r1525324;
        double r1525326 = r1525324 + r1525325;
        double r1525327 = r1525323 / r1525326;
        return r1525327;
}

Error

Bits error versus x

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Results

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Target

Original15.0
Target0.1
Herbie0.1
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Initial program 15.0

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

  2. Simplified15.0

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

  4. Applied add-sqr-sqrt15.0

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

  5. Applied associate-/r*14.9

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

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

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

  8. Applied associate-/l*15.0

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

  9. Simplified0.1

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

  10. Taylor expanded around inf 0.1

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

  11. Final simplification0.1

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

Reproduce

herbie shell --seed 2019130 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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