simple fma test

Average Error: 44.8 → 7.6

Time: 16.2s

Precision: 64

Math

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

↓

\[\log \left(\frac{e^{\left(\left(\mathsf{fma}\left(x, y, z\right) - \left(z + x \cdot y\right)\right)\right)}}{e}\right)\]

C

double f(double x, double y, double z) {
        double r1379168 = x;
        double r1379169 = y;
        double r1379170 = z;
        double r1379171 = fma(r1379168, r1379169, r1379170);
        double r1379172 = 1.0;
        double r1379173 = r1379168 * r1379169;
        double r1379174 = r1379173 + r1379170;
        double r1379175 = r1379172 + r1379174;
        double r1379176 = r1379171 - r1379175;
        return r1379176;
}

↓

double f(double x, double y, double z) {
        double r1379177 = x;
        double r1379178 = y;
        double r1379179 = z;
        double r1379180 = fma(r1379177, r1379178, r1379179);
        double r1379181 = r1379177 * r1379178;
        double r1379182 = r1379179 + r1379181;
        double r1379183 = r1379180 - r1379182;
        double r1379184 = /* ERROR: no posit support in C */;
        double r1379185 = /* ERROR: no posit support in C */;
        double r1379186 = exp(r1379185);
        double r1379187 = exp(1.0);
        double r1379188 = r1379186 / r1379187;
        double r1379189 = log(r1379188);
        return r1379189;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	44.8
Target	0
Herbie	7.6

\[-1\]

Derivation

1. Initial program 44.8

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

3. Applied add-log-exp 46.8

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

4. Applied add-log-exp 46.8

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

5. Applied sum-log 46.9

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

6. Applied add-log-exp 47.3

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

7. Applied diff-log 47.3

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

8. Simplified 30.7

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

10. Applied add-log-exp 31.7

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

11. Applied add-log-exp 31.9

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

12. Applied diff-log 31.9

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

13. Simplified 14.8

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

14. Taylor expanded around -inf 7.5

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

16. Applied insert-posit16 7.6

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

17. Final simplification 7.6

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

Reproduce

herbie shell --seed 2019151 
(FPCore (x y z)
  :name "simple fma test"

  :herbie-target
  -1

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