Average Error: 0.0 → 1.0
Time: 1.5s
Precision: 64
\[\left(x + y\right) \cdot \left(1 - z\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -0.002700652192797984 \lor \neg \left(z \le 22132.784332979267\right):\\ \;\;\;\;\mathsf{fma}\left(1, y, -\mathsf{fma}\left(x, z, z \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, -z, \mathsf{fma}\left(1, x, 1 \cdot y\right)\right)\\ \end{array}\]
\left(x + y\right) \cdot \left(1 - z\right)
↓
\begin{array}{l}
\mathbf{if}\;z \le -0.002700652192797984 \lor \neg \left(z \le 22132.784332979267\right):\\
\;\;\;\;\mathsf{fma}\left(1, y, -\mathsf{fma}\left(x, z, z \cdot y\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(x, -z, \mathsf{fma}\left(1, x, 1 \cdot y\right)\right)\\

\end{array}
double code(double x, double y, double z) {
	return ((x + y) * (1.0 - z));
}
↓
double code(double x, double y, double z) {
	double VAR;
	if (((z <= -0.0027006521927979837) || !(z <= 22132.784332979267))) {
		VAR = fma(1.0, y, -fma(x, z, (z * y)));
	} else {
		VAR = fma(x, -z, fma(1.0, x, (1.0 * y)));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if z < -0.0027006521927979837 or 22132.784332979267 < z

    1. Initial program 0.0

      \[\left(x + y\right) \cdot \left(1 - z\right)\]

    2. Taylor expanded around inf 1.1

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

    3. Simplified1.1

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

    if -0.0027006521927979837 < z < 22132.784332979267

    1. Initial program 0.0

      \[\left(x + y\right) \cdot \left(1 - z\right)\]
    2. Using strategy rm

    3. Applied sub-neg0.0

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

    4. Applied distribute-lft-in0.0

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

    5. Simplified0.0

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

    6. Simplified0.0

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

    7. Taylor expanded around 0 0.9

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

    8. Simplified0.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, -z, \mathsf{fma}\left(1, x, 1 \cdot y\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -0.002700652192797984 \lor \neg \left(z \le 22132.784332979267\right):\\ \;\;\;\;\mathsf{fma}\left(1, y, -\mathsf{fma}\left(x, z, z \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, -z, \mathsf{fma}\left(1, x, 1 \cdot y\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020105 +o rules:numerics
(FPCore (x y z)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, H"
  :precision binary64
  (* (+ x y) (- 1 z)))