Average Error: 0.0 → 0.0
Time: 15.6s
Precision: 64
\[\frac{x - y}{x + y}\]
↓
\[\frac{{\left(\frac{x}{x + y}\right)}^{3} - {\left(\frac{y}{x + y}\right)}^{3}}{\mathsf{fma}\left(\frac{x}{x + y}, \frac{x}{x + y}, \frac{y}{x + y} \cdot \left(\frac{x}{x + y} + \frac{y}{x + y}\right)\right)}\]
\frac{x - y}{x + y}
↓
\frac{{\left(\frac{x}{x + y}\right)}^{3} - {\left(\frac{y}{x + y}\right)}^{3}}{\mathsf{fma}\left(\frac{x}{x + y}, \frac{x}{x + y}, \frac{y}{x + y} \cdot \left(\frac{x}{x + y} + \frac{y}{x + y}\right)\right)}
double f(double x, double y) {
        double r450386 = x;
        double r450387 = y;
        double r450388 = r450386 - r450387;
        double r450389 = r450386 + r450387;
        double r450390 = r450388 / r450389;
        return r450390;
}
↓
double f(double x, double y) {
        double r450391 = x;
        double r450392 = y;
        double r450393 = r450391 + r450392;
        double r450394 = r450391 / r450393;
        double r450395 = 3.0;
        double r450396 = pow(r450394, r450395);
        double r450397 = r450392 / r450393;
        double r450398 = pow(r450397, r450395);
        double r450399 = r450396 - r450398;
        double r450400 = r450394 + r450397;
        double r450401 = r450397 * r450400;
        double r450402 = fma(r450394, r450394, r450401);
        double r450403 = r450399 / r450402;
        return r450403;
}

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. Using strategy rm

  3. Applied div-sub0.0

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

  5. Applied flip3--0.0

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

  6. Simplified0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019322 +o rules:numerics
(FPCore (x y)
  :name "Data.Colour.RGB:hslsv from colour-2.3.3, D"
  :precision binary64

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

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