Average Error: 45.2 → 45.1
Time: 5.9s
Precision: 64
\[\mathsf{fma}\left(x, y, z\right) - \left(1 + \left(x \cdot y + z\right)\right)\]
↓
\[\left(\mathsf{fma}\left(x, y, z\right) - \left(1 + \frac{x \cdot y}{\frac{x \cdot y - z}{x \cdot y}}\right)\right) + \frac{z}{\frac{x \cdot y - z}{z}}\]
\mathsf{fma}\left(x, y, z\right) - \left(1 + \left(x \cdot y + z\right)\right)
↓
\left(\mathsf{fma}\left(x, y, z\right) - \left(1 + \frac{x \cdot y}{\frac{x \cdot y - z}{x \cdot y}}\right)\right) + \frac{z}{\frac{x \cdot y - z}{z}}
double f(double x, double y, double z) {
        double r70996 = x;
        double r70997 = y;
        double r70998 = z;
        double r70999 = fma(r70996, r70997, r70998);
        double r71000 = 1.0;
        double r71001 = r70996 * r70997;
        double r71002 = r71001 + r70998;
        double r71003 = r71000 + r71002;
        double r71004 = r70999 - r71003;
        return r71004;
}
↓
double f(double x, double y, double z) {
        double r71005 = x;
        double r71006 = y;
        double r71007 = z;
        double r71008 = fma(r71005, r71006, r71007);
        double r71009 = 1.0;
        double r71010 = r71005 * r71006;
        double r71011 = r71010 - r71007;
        double r71012 = r71011 / r71010;
        double r71013 = r71010 / r71012;
        double r71014 = r71009 + r71013;
        double r71015 = r71008 - r71014;
        double r71016 = r71011 / r71007;
        double r71017 = r71007 / r71016;
        double r71018 = r71015 + r71017;
        return r71018;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original45.2
Target0
Herbie45.1
\[-1\]

Derivation

  1. Initial program 45.2

    \[\mathsf{fma}\left(x, y, z\right) - \left(1 + \left(x \cdot y + z\right)\right)\]
  2. Using strategy rm

  3. Applied flip-+45.9

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

  5. Applied div-sub45.9

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

  6. Applied associate-+r-45.9

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

  7. Applied associate--r-45.9

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

  9. Applied associate-/l*45.6

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

  11. Applied associate-/l*45.1

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

  12. Final simplification45.1

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

Reproduce

herbie shell --seed 2020001 
(FPCore (x y z)
  :name "simple fma test"
  :precision binary64

  :herbie-target
  -1

  (- (fma x y z) (+ 1 (+ (* x y) z))))