Average Error: 45.0 → 45.0
Time: 1.9s
Precision: binary64
\[\mathsf{fma}\left(x, y, z\right) - \left(1 + \left(x \cdot y + z\right)\right)\]
↓
\[\mathsf{fma}\left(x, y, z\right) - \left(1 + \left(z + x \cdot y\right)\right)\]
\mathsf{fma}\left(x, y, z\right) - \left(1 + \left(x \cdot y + z\right)\right)
↓
\mathsf{fma}\left(x, y, z\right) - \left(1 + \left(z + x \cdot y\right)\right)
(FPCore (x y z) :precision binary64 (- (fma x y z) (+ 1.0 (+ (* x y) z))))
↓
(FPCore (x y z) :precision binary64 (- (fma x y z) (+ 1.0 (+ z (* x y)))))
double code(double x, double y, double z) {
	return ((double) (((double) fma(x, y, z)) - ((double) (1.0 + ((double) (((double) (x * y)) + z))))));
}
↓
double code(double x, double y, double z) {
	return ((double) (((double) fma(x, y, z)) - ((double) (1.0 + ((double) (z + ((double) (x * y))))))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original45.0
Target0
Herbie45.0
\[-1\]

Derivation

  1. Initial program Error: 45.0 bits

    \[\mathsf{fma}\left(x, y, z\right) - \left(1 + \left(x \cdot y + z\right)\right)\]

  2. Final simplificationError: 45.0 bits

    \[\leadsto \mathsf{fma}\left(x, y, z\right) - \left(1 + \left(z + x \cdot y\right)\right)\]

Reproduce

herbie shell --seed 2020204 
(FPCore (x y z)
  :name "simple fma test"
  :precision binary64

  :herbie-target
  -1.0

  (- (fma x y z) (+ 1.0 (+ (* x y) z))))