Average Error: 0.0 → 0.0
Time: 2.0s
Precision: 64
\[\left(\frac{x}{2} + y \cdot x\right) + z\]
↓
\[\mathsf{fma}\left(x, 0.5 + y, z\right)\]
\left(\frac{x}{2} + y \cdot x\right) + z
↓
\mathsf{fma}\left(x, 0.5 + y, z\right)
double f(double x, double y, double z) {
        double r282266 = x;
        double r282267 = 2.0;
        double r282268 = r282266 / r282267;
        double r282269 = y;
        double r282270 = r282269 * r282266;
        double r282271 = r282268 + r282270;
        double r282272 = z;
        double r282273 = r282271 + r282272;
        return r282273;
}
↓
double f(double x, double y, double z) {
        double r282274 = x;
        double r282275 = 0.5;
        double r282276 = y;
        double r282277 = r282275 + r282276;
        double r282278 = z;
        double r282279 = fma(r282274, r282277, r282278);
        return r282279;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.0

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

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, \frac{x}{2}\right) + z}\]

  3. Taylor expanded around 0 0.0

    \[\leadsto \color{blue}{0.5 \cdot x + \left(z + x \cdot y\right)}\]

  4. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.5 + y, z\right)}\]

  5. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(x, 0.5 + y, z\right)\]

Reproduce

herbie shell --seed 2020043 +o rules:numerics
(FPCore (x y z)
  :name "Data.Histogram.Bin.BinF:$cfromIndex from histogram-fill-0.8.4.1"
  :precision binary64
  (+ (+ (/ x 2) (* y x)) z))