Average Error: 45.1 → 25.6
Time: 6.4s
Precision: binary64
\[\mathsf{fma}\left(x, y, z\right) - \left(1 + \left(x \cdot y + z\right)\right) \]
↓
\[\left(\mathsf{fma}\left(x, y, z\right) - z\right) - 1 \]
\mathsf{fma}\left(x, y, z\right) - \left(1 + \left(x \cdot y + z\right)\right)
↓
\left(\mathsf{fma}\left(x, y, z\right) - z\right) - 1

(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) z) 1.0))
double code(double x, double y, double z) {
	return fma(x, y, z) - (1.0 + ((x * y) + z));
}
↓
double code(double x, double y, double z) {
	return (fma(x, y, z) - z) - 1.0;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original45.1
Target0
Herbie25.6
\[-1 \]

Derivation

  1. Initial program 45.1

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

  2. Taylor expanded around 0 46.2

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

  3. Simplified46.2

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, z\right) - \left(z + 1\right)} \]
  4. Using strategy rm

  5. Applied associate--r+_binary6425.6

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

  6. Final simplification25.6

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

Reproduce

herbie shell --seed 2021198 
(FPCore (x y z)
  :name "simple fma test"
  :precision binary64

  :herbie-target
  -1.0

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