Average Error: 0.0 → 0.0
Time: 13.5s
Precision: 64
\[\frac{x - y}{x + y}\]
↓
\[\frac{\mathsf{fma}\left(\frac{x}{y + x}, \frac{x}{y + x}, \frac{y}{y + x} \cdot \left(-\frac{y}{y + x}\right)\right)}{\frac{x}{y + x} + \frac{y}{y + x}}\]
\frac{x - y}{x + y}
↓
\frac{\mathsf{fma}\left(\frac{x}{y + x}, \frac{x}{y + x}, \frac{y}{y + x} \cdot \left(-\frac{y}{y + x}\right)\right)}{\frac{x}{y + x} + \frac{y}{y + x}}
double f(double x, double y) {
        double r687088 = x;
        double r687089 = y;
        double r687090 = r687088 - r687089;
        double r687091 = r687088 + r687089;
        double r687092 = r687090 / r687091;
        return r687092;
}
↓
double f(double x, double y) {
        double r687093 = x;
        double r687094 = y;
        double r687095 = r687094 + r687093;
        double r687096 = r687093 / r687095;
        double r687097 = r687094 / r687095;
        double r687098 = -r687097;
        double r687099 = r687097 * r687098;
        double r687100 = fma(r687096, r687096, r687099);
        double r687101 = r687096 + r687097;
        double r687102 = r687100 / r687101;
        return r687102;
}

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 flip--0.0

    \[\leadsto \color{blue}{\frac{\frac{x}{x + y} \cdot \frac{x}{x + y} - \frac{y}{x + y} \cdot \frac{y}{x + y}}{\frac{x}{x + y} + \frac{y}{x + y}}}\]

  6. Simplified0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019194 +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)))