Average Error: 0.0 → 0.0
Time: 3.0s
Precision: binary64
\[x + y \cdot \left(z + x\right) \]
↓
\[\mathsf{fma}\left(x, y, \mathsf{fma}\left(y, z, x\right)\right) \]
x + y \cdot \left(z + x\right)
↓
\mathsf{fma}\left(x, y, \mathsf{fma}\left(y, z, x\right)\right)

(FPCore (x y z) :precision binary64 (+ x (* y (+ z x))))
↓
(FPCore (x y z) :precision binary64 (fma x y (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 fma(x, y, 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. Applied egg-rr41.2

    \[\leadsto \color{blue}{\frac{{\left(y \cdot \left(x + z\right)\right)}^{3} + {x}^{3}}{\left(y \cdot \left(x + z\right)\right) \cdot \left(y \cdot \left(x + z\right)\right) + \left(x \cdot x - \left(y \cdot \left(x + z\right)\right) \cdot x\right)}} \]

  4. Taylor expanded in y around 0 0.0

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

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2022125 
(FPCore (x y z)
  :name "Main:bigenough2 from A"
  :precision binary64
  (+ x (* y (+ z x))))