Average Error: 0.0 → 0.0
Time: 7.5s
Precision: 64
\[\frac{x - y}{x + y}\]
↓
\[\frac{{\left(\frac{x}{y + x}\right)}^{3} - {\left(\frac{y}{y + x}\right)}^{3}}{\mathsf{fma}\left(\frac{y}{y + x}, \frac{x}{y + x} + \frac{y}{y + x}, \frac{x}{y + x} \cdot \frac{x}{y + x}\right)}\]
\frac{x - y}{x + y}
↓
\frac{{\left(\frac{x}{y + x}\right)}^{3} - {\left(\frac{y}{y + x}\right)}^{3}}{\mathsf{fma}\left(\frac{y}{y + x}, \frac{x}{y + x} + \frac{y}{y + x}, \frac{x}{y + x} \cdot \frac{x}{y + x}\right)}
double f(double x, double y) {
        double r590980 = x;
        double r590981 = y;
        double r590982 = r590980 - r590981;
        double r590983 = r590980 + r590981;
        double r590984 = r590982 / r590983;
        return r590984;
}
↓
double f(double x, double y) {
        double r590985 = x;
        double r590986 = y;
        double r590987 = r590986 + r590985;
        double r590988 = r590985 / r590987;
        double r590989 = 3.0;
        double r590990 = pow(r590988, r590989);
        double r590991 = r590986 / r590987;
        double r590992 = pow(r590991, r590989);
        double r590993 = r590990 - r590992;
        double r590994 = r590988 + r590991;
        double r590995 = r590988 * r590988;
        double r590996 = fma(r590991, r590994, r590995);
        double r590997 = r590993 / r590996;
        return r590997;
}

Error

Bits error versus x

Bits error versus y

Target

Original0.0
Target0.0
Herbie0.0
\[\frac{x}{x + y} - \frac{y}{x + y}\]

Derivation

  1. Initial program 0.0

    \[\frac{x - y}{x + y}\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\frac{x - y}{y + x}}\]
  3. Using strategy rm

  4. Applied div-sub0.0

    \[\leadsto \color{blue}{\frac{x}{y + x} - \frac{y}{y + x}}\]
  5. Using strategy rm

  6. Applied flip3--0.0

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

  7. Simplified0.0

    \[\leadsto \frac{{\left(\frac{x}{y + x}\right)}^{3} - {\left(\frac{y}{y + x}\right)}^{3}}{\color{blue}{\mathsf{fma}\left(\frac{y}{y + x}, \frac{y}{y + x} + \frac{x}{y + x}, \frac{x}{y + x} \cdot \frac{x}{y + x}\right)}}\]

  8. Final simplification0.0

    \[\leadsto \frac{{\left(\frac{x}{y + x}\right)}^{3} - {\left(\frac{y}{y + x}\right)}^{3}}{\mathsf{fma}\left(\frac{y}{y + x}, \frac{x}{y + x} + \frac{y}{y + x}, \frac{x}{y + x} \cdot \frac{x}{y + x}\right)}\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x y)
  :name "Data.Colour.RGB:hslsv from colour-2.3.3, D"

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

  (/ (- x y) (+ x y)))