x / (x^2 + 1)

Average Error: 14.7 → 0.0

Time: 1.2m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -61349745.21847331:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{\left(x \cdot x\right) \cdot x}\\ \mathbf{elif}\;x \le 405.32013382536394:\\ \;\;\;\;\frac{x}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} \cdot \frac{1}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{\left(x \cdot x\right) \cdot x}\\ \end{array}\]

C

double f(double x) {
        double r9565955 = x;
        double r9565956 = r9565955 * r9565955;
        double r9565957 = 1.0;
        double r9565958 = r9565956 + r9565957;
        double r9565959 = r9565955 / r9565958;
        return r9565959;
}

↓

double f(double x) {
        double r9565960 = x;
        double r9565961 = -61349745.21847331;
        bool r9565962 = r9565960 <= r9565961;
        double r9565963 = 1.0;
        double r9565964 = 5.0;
        double r9565965 = pow(r9565960, r9565964);
        double r9565966 = r9565963 / r9565965;
        double r9565967 = r9565963 / r9565960;
        double r9565968 = r9565966 + r9565967;
        double r9565969 = r9565960 * r9565960;
        double r9565970 = r9565969 * r9565960;
        double r9565971 = r9565963 / r9565970;
        double r9565972 = r9565968 - r9565971;
        double r9565973 = 405.32013382536394;
        bool r9565974 = r9565960 <= r9565973;
        double r9565975 = fma(r9565960, r9565960, r9565963);
        double r9565976 = sqrt(r9565975);
        double r9565977 = r9565960 / r9565976;
        double r9565978 = r9565963 / r9565976;
        double r9565979 = r9565977 * r9565978;
        double r9565980 = r9565974 ? r9565979 : r9565972;
        double r9565981 = r9565962 ? r9565972 : r9565980;
        return r9565981;
}

Error

Bits error versus x

Target

Original	14.7
Target	0.1
Herbie	0.0

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

Derivation

1. Split input into 2 regimes

2. if x < -61349745.21847331 or 405.32013382536394 < x

   1. Initial program 30.1

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

   2. Simplified 30.1

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

   3. Taylor expanded around inf 0.0

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

   4. Simplified 0.0

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

   if -61349745.21847331 < x < 405.32013382536394

   1. Initial program 0.0

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

   2. Simplified 0.0

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

   4. Applied add-sqr-sqrt 0.0

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

   5. Applied *-un-lft-identity 0.0

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

   6. Applied times-frac 0.0

      \[\leadsto \color{blue}{\frac{1}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} \cdot \frac{x}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -61349745.21847331:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{\left(x \cdot x\right) \cdot x}\\ \mathbf{elif}\;x \le 405.32013382536394:\\ \;\;\;\;\frac{x}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} \cdot \frac{1}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{\left(x \cdot x\right) \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019125 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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