simple fma test

Average Error: 45.1 → 31.4

Time: 15.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le 8.355733002062013515976805283223099997005 \cdot 10^{99}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(x, y, z\right) - \frac{x}{1 - \frac{z}{x \cdot y}} \cdot y\right) - 1\right) + \frac{z \cdot z}{x \cdot y - z}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(x, y, z\right) - \frac{x}{\frac{1 - \frac{z}{x \cdot y}}{y}}\right) - 1\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r51369 = x;
        double r51370 = y;
        double r51371 = z;
        double r51372 = fma(r51369, r51370, r51371);
        double r51373 = 1.0;
        double r51374 = r51369 * r51370;
        double r51375 = r51374 + r51371;
        double r51376 = r51373 + r51375;
        double r51377 = r51372 - r51376;
        return r51377;
}

↓

double f(double x, double y, double z) {
        double r51378 = z;
        double r51379 = 8.355733002062014e+99;
        bool r51380 = r51378 <= r51379;
        double r51381 = x;
        double r51382 = y;
        double r51383 = fma(r51381, r51382, r51378);
        double r51384 = 1.0;
        double r51385 = r51381 * r51382;
        double r51386 = r51378 / r51385;
        double r51387 = r51384 - r51386;
        double r51388 = r51381 / r51387;
        double r51389 = r51388 * r51382;
        double r51390 = r51383 - r51389;
        double r51391 = 1.0;
        double r51392 = r51390 - r51391;
        double r51393 = r51378 * r51378;
        double r51394 = r51385 - r51378;
        double r51395 = r51393 / r51394;
        double r51396 = r51392 + r51395;
        double r51397 = r51387 / r51382;
        double r51398 = r51381 / r51397;
        double r51399 = r51383 - r51398;
        double r51400 = r51399 - r51391;
        double r51401 = r51380 ? r51396 : r51400;
        return r51401;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	45.1
Target	0
Herbie	31.4

\[-1\]

Derivation

1. Split input into 2 regimes

2. if z < 8.355733002062014e+99

   1. Initial program 41.8

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

   3. Applied flip-+ 42.4

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

   5. Applied div-sub 42.4

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

   6. Applied associate-+r- 42.4

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

   7. Applied associate--r- 42.4

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

   8. Simplified 30.7

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

   10. Applied associate-/r/ 26.1

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

   if 8.355733002062014e+99 < z

   1. Initial program 62.2

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

   3. Applied flip-+ 63.7

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

   5. Applied div-sub 63.7

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

   6. Applied associate-+r- 63.7

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

   7. Applied associate--r- 63.7

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

   8. Simplified 63.3

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

   9. Taylor expanded around 0 59.3

      \[\leadsto \left(\left(\mathsf{fma}\left(x, y, z\right) - \frac{x}{\frac{1 - \frac{z}{x \cdot y}}{y}}\right) - 1\right) + \color{blue}{0}\]
3. Recombined 2 regimes into one program.

4. Final simplification 31.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le 8.355733002062013515976805283223099997005 \cdot 10^{99}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(x, y, z\right) - \frac{x}{1 - \frac{z}{x \cdot y}} \cdot y\right) - 1\right) + \frac{z \cdot z}{x \cdot y - z}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(x, y, z\right) - \frac{x}{\frac{1 - \frac{z}{x \cdot y}}{y}}\right) - 1\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(FPCore (x y z)
  :name "simple fma test"
  :precision binary64

  :herbie-target
  -1

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