Average Error: 0.0 → 0.0
Time: 5.7s
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 r8243262 = x;
        double r8243263 = 2.0;
        double r8243264 = r8243262 / r8243263;
        double r8243265 = y;
        double r8243266 = r8243265 * r8243262;
        double r8243267 = r8243264 + r8243266;
        double r8243268 = z;
        double r8243269 = r8243267 + r8243268;
        return r8243269;
}
↓
double f(double x, double y, double z) {
        double r8243270 = x;
        double r8243271 = 0.5;
        double r8243272 = y;
        double r8243273 = r8243271 + r8243272;
        double r8243274 = z;
        double r8243275 = fma(r8243270, r8243273, r8243274);
        return r8243275;
}

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

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

  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 2019169 +o rules:numerics
(FPCore (x y z)
  :name "Data.Histogram.Bin.BinF:$cfromIndex from histogram-fill-0.8.4.1"
  (+ (+ (/ x 2.0) (* y x)) z))