Average Error: 0.0 → 0.0
Time: 1.8s
Precision: binary64
\[\left(\frac{x}{2} + y \cdot x\right) + z \]
↓
\[\mathsf{fma}\left(y, x, z + x \cdot 0.5\right) \]
\left(\frac{x}{2} + y \cdot x\right) + z
↓
\mathsf{fma}\left(y, x, z + x \cdot 0.5\right)

(FPCore (x y z) :precision binary64 (+ (+ (/ x 2.0) (* y x)) z))
↓
(FPCore (x y z) :precision binary64 (fma y x (+ z (* x 0.5))))
double code(double x, double y, double z) {
	return ((x / 2.0) + (y * x)) + z;
}
↓
double code(double x, double y, double z) {
	return fma(y, x, (z + (x * 0.5)));
}

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 in x around 0 0.0

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

  4. Applied fma-def_binary640.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2022067 
(FPCore (x y z)
  :name "Data.Histogram.Bin.BinF:$cfromIndex from histogram-fill-0.8.4.1"
  :precision binary64
  (+ (+ (/ x 2.0) (* y x)) z))