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

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

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

  3. Taylor expanded in y around 0 0.0

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

  4. Applied fma-def_binary640.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2022067 
(FPCore (x y z)
  :name "SynthBasics:oscSampleBasedAux from YampaSynth-0.2"
  :precision binary64
  (+ x (* y (- z x))))