Average Error: 0.0 → 0.0
Time: 4.1s
Precision: 64
\[\frac{x - y}{x + y}\]
↓
\[\frac{{\left(\frac{x}{x + y}\right)}^{3} - {\left(\frac{y}{x + y}\right)}^{3}}{\mathsf{fma}\left(\frac{y}{x + y}, \frac{x}{x + y} + \frac{y}{x + y}, \frac{x}{x + y} \cdot \frac{x}{x + y}\right)}\]
\frac{x - y}{x + y}
↓
\frac{{\left(\frac{x}{x + y}\right)}^{3} - {\left(\frac{y}{x + y}\right)}^{3}}{\mathsf{fma}\left(\frac{y}{x + y}, \frac{x}{x + y} + \frac{y}{x + y}, \frac{x}{x + y} \cdot \frac{x}{x + y}\right)}
double f(double x, double y) {
        double r1000466 = x;
        double r1000467 = y;
        double r1000468 = r1000466 - r1000467;
        double r1000469 = r1000466 + r1000467;
        double r1000470 = r1000468 / r1000469;
        return r1000470;
}
↓
double f(double x, double y) {
        double r1000471 = x;
        double r1000472 = y;
        double r1000473 = r1000471 + r1000472;
        double r1000474 = r1000471 / r1000473;
        double r1000475 = 3.0;
        double r1000476 = pow(r1000474, r1000475);
        double r1000477 = r1000472 / r1000473;
        double r1000478 = pow(r1000477, r1000475);
        double r1000479 = r1000476 - r1000478;
        double r1000480 = r1000474 + r1000477;
        double r1000481 = r1000474 * r1000474;
        double r1000482 = fma(r1000477, r1000480, r1000481);
        double r1000483 = r1000479 / r1000482;
        return r1000483;
}

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

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2020100 +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)))