(+ (+ (* (* y xz) xz) (* x xz)) z)

Average Error: 0.1 → 0.1

Time: 1.5s

Precision: binary64

Math

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

↓

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

C

double code(double y, double xz, double x, double z) {
	return ((double) (((double) (((double) (((double) (y * xz)) * xz)) + ((double) (x * xz)))) + z));
}

↓

double code(double y, double xz, double x, double z) {
	return ((double) (z + ((double) (xz * ((double) (((double) (y * xz)) + x))))));
}

Error

Bits error versus y

Bits error versus xz

Bits error versus x

Bits error versus z

Derivation

1. Initial program 0.1

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

2. Simplified 0.1

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

3. Final simplification 0.1

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

Reproduce

herbie shell --seed 2020153 
(FPCore (y xz x z)
  :name "(+ (+ (* (* y xz) xz) (* x xz)) z)"
  :precision binary64
  (+ (+ (* (* y xz) xz) (* x xz)) z))