Average Error: 45.2 → 45.1
Time: 6.1s
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 r96496 = x;
        double r96497 = y;
        double r96498 = z;
        double r96499 = fma(r96496, r96497, r96498);
        double r96500 = 1.0;
        double r96501 = r96496 * r96497;
        double r96502 = r96501 + r96498;
        double r96503 = r96500 + r96502;
        double r96504 = r96499 - r96503;
        return r96504;
}
↓
double f(double x, double y, double z) {
        double r96505 = x;
        double r96506 = y;
        double r96507 = z;
        double r96508 = fma(r96505, r96506, r96507);
        double r96509 = 1.0;
        double r96510 = r96505 * r96506;
        double r96511 = r96510 - r96507;
        double r96512 = r96511 / r96510;
        double r96513 = r96510 / r96512;
        double r96514 = r96509 + r96513;
        double r96515 = r96508 - r96514;
        double r96516 = r96511 / r96507;
        double r96517 = r96507 / r96516;
        double r96518 = r96515 + r96517;
        return r96518;
}

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))))