Average Error: 0.0 → 0.0
Time: 2.6s
Precision: 64
\[\left(\frac{x}{2} + y \cdot x\right) + z\]
↓
\[\frac{\frac{x}{2} - y \cdot x}{\frac{\frac{x}{2} - y \cdot x}{\mathsf{fma}\left(x, y, \frac{x}{2}\right)}} + z\]
\left(\frac{x}{2} + y \cdot x\right) + z
↓
\frac{\frac{x}{2} - y \cdot x}{\frac{\frac{x}{2} - y \cdot x}{\mathsf{fma}\left(x, y, \frac{x}{2}\right)}} + z
double f(double x, double y, double z) {
        double r272270 = x;
        double r272271 = 2.0;
        double r272272 = r272270 / r272271;
        double r272273 = y;
        double r272274 = r272273 * r272270;
        double r272275 = r272272 + r272274;
        double r272276 = z;
        double r272277 = r272275 + r272276;
        return r272277;
}
↓
double f(double x, double y, double z) {
        double r272278 = x;
        double r272279 = 2.0;
        double r272280 = r272278 / r272279;
        double r272281 = y;
        double r272282 = r272281 * r272278;
        double r272283 = r272280 - r272282;
        double r272284 = fma(r272278, r272281, r272280);
        double r272285 = r272283 / r272284;
        double r272286 = r272283 / r272285;
        double r272287 = z;
        double r272288 = r272286 + r272287;
        return r272288;
}

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. Using strategy rm

  3. Applied flip-+20.4

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

  4. Simplified20.4

    \[\leadsto \frac{\color{blue}{\left(\frac{x}{2} - y \cdot x\right) \cdot \mathsf{fma}\left(x, y, \frac{x}{2}\right)}}{\frac{x}{2} - y \cdot x} + z\]
  5. Using strategy rm

  6. Applied associate-/l*0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2020062 +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))