Average Error: 0.0 → 0.0
Time: 1.7s
Precision: binary64
\[x + y \cdot \left(z + x\right) \]
↓
\[y \cdot x + \mathsf{fma}\left(y, z, x\right) \]
x + y \cdot \left(z + x\right)
↓
y \cdot x + \mathsf{fma}\left(y, z, x\right)

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

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.0

    \[x + y \cdot \left(z + x\right) \]

  2. Simplified0.0

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

  3. Taylor expanded in y around 0 0.0

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

  4. Applied fma-def_binary640.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2022067 
(FPCore (x y z)
  :name "Main:bigenough2 from A"
  :precision binary64
  (+ x (* y (+ z x))))