Average Error: 15.6 → 0.0
Time: 1.0s
Precision: 64
\[\frac{x + y}{\left(x \cdot 2\right) \cdot y}\]
↓
\[\mathsf{fma}\left(0.5, \frac{1}{y}, 0.5 \cdot \frac{1}{x}\right)\]
\frac{x + y}{\left(x \cdot 2\right) \cdot y}
↓
\mathsf{fma}\left(0.5, \frac{1}{y}, 0.5 \cdot \frac{1}{x}\right)
double f(double x, double y) {
        double r513608 = x;
        double r513609 = y;
        double r513610 = r513608 + r513609;
        double r513611 = 2.0;
        double r513612 = r513608 * r513611;
        double r513613 = r513612 * r513609;
        double r513614 = r513610 / r513613;
        return r513614;
}
↓
double f(double x, double y) {
        double r513615 = 0.5;
        double r513616 = 1.0;
        double r513617 = y;
        double r513618 = r513616 / r513617;
        double r513619 = x;
        double r513620 = r513616 / r513619;
        double r513621 = r513615 * r513620;
        double r513622 = fma(r513615, r513618, r513621);
        return r513622;
}

Error

Bits error versus x

Bits error versus y

Target

Original15.6
Target0.0
Herbie0.0
\[\frac{0.5}{x} + \frac{0.5}{y}\]

Derivation

  1. Initial program 15.6

    \[\frac{x + y}{\left(x \cdot 2\right) \cdot y}\]

  2. Taylor expanded around 0 0.0

    \[\leadsto \color{blue}{0.5 \cdot \frac{1}{y} + 0.5 \cdot \frac{1}{x}}\]

  3. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{1}{y}, 0.5 \cdot \frac{1}{x}\right)}\]

  4. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(0.5, \frac{1}{y}, 0.5 \cdot \frac{1}{x}\right)\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (x y)
  :name "Linear.Projection:inversePerspective from linear-1.19.1.3, C"
  :precision binary64

  :herbie-target
  (+ (/ 0.5 x) (/ 0.5 y))

  (/ (+ x y) (* (* x 2) y)))