Average Error: 1.6 → 1.6
Time: 11.4s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\left|\mathsf{fma}\left(\frac{x}{y}, 1 - z, \frac{4}{y}\right)\right|\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\left|\mathsf{fma}\left(\frac{x}{y}, 1 - z, \frac{4}{y}\right)\right|
double f(double x, double y, double z) {
        double r36184 = x;
        double r36185 = 4.0;
        double r36186 = r36184 + r36185;
        double r36187 = y;
        double r36188 = r36186 / r36187;
        double r36189 = r36184 / r36187;
        double r36190 = z;
        double r36191 = r36189 * r36190;
        double r36192 = r36188 - r36191;
        double r36193 = fabs(r36192);
        return r36193;
}
↓
double f(double x, double y, double z) {
        double r36194 = x;
        double r36195 = y;
        double r36196 = r36194 / r36195;
        double r36197 = 1.0;
        double r36198 = z;
        double r36199 = r36197 - r36198;
        double r36200 = 4.0;
        double r36201 = r36200 / r36195;
        double r36202 = fma(r36196, r36199, r36201);
        double r36203 = fabs(r36202);
        return r36203;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 1.6

    \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]

  2. Taylor expanded around 0 3.4

    \[\leadsto \left|\color{blue}{\left(4 \cdot \frac{1}{y} + \frac{x}{y}\right) - \frac{x \cdot z}{y}}\right|\]

  3. Simplified1.6

    \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\frac{x}{y}, 1 - z, \frac{4}{y}\right)}\right|\]

  4. Final simplification1.6

    \[\leadsto \left|\mathsf{fma}\left(\frac{x}{y}, 1 - z, \frac{4}{y}\right)\right|\]

Reproduce

herbie shell --seed 2019305 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))