simple fma test

Average Error: 45.0 → 31.1

Time: 22.3s

Precision: 64

Math

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

↓

\[\log \left(e^{\left(\sqrt[3]{\sqrt[3]{\left(\left(\left(\mathsf{fma}\left(x, y, z\right) - z\right) - \left(1 + x \cdot y\right)\right) \cdot \left(\left(\mathsf{fma}\left(x, y, z\right) - z\right) - \left(1 + x \cdot y\right)\right)\right) \cdot \left(\left(\mathsf{fma}\left(x, y, z\right) - z\right) - \left(1 + x \cdot y\right)\right)}} \cdot \sqrt[3]{\sqrt[3]{\left(\left(\left(\mathsf{fma}\left(x, y, z\right) - z\right) - \left(1 + x \cdot y\right)\right) \cdot \left(\left(\mathsf{fma}\left(x, y, z\right) - z\right) - \left(1 + x \cdot y\right)\right)\right) \cdot \left(\left(\mathsf{fma}\left(x, y, z\right) - z\right) - \left(1 + x \cdot y\right)\right)}}\right) \cdot \sqrt[3]{\sqrt[3]{\left(\left(\left(\mathsf{fma}\left(x, y, z\right) - z\right) - \left(1 + x \cdot y\right)\right) \cdot \left(\left(\mathsf{fma}\left(x, y, z\right) - z\right) - \left(1 + x \cdot y\right)\right)\right) \cdot \left(\left(\mathsf{fma}\left(x, y, z\right) - z\right) - \left(1 + x \cdot y\right)\right)}}}\right)\]

C

double f(double x, double y, double z) {
        double r4852754 = x;
        double r4852755 = y;
        double r4852756 = z;
        double r4852757 = fma(r4852754, r4852755, r4852756);
        double r4852758 = 1.0;
        double r4852759 = r4852754 * r4852755;
        double r4852760 = r4852759 + r4852756;
        double r4852761 = r4852758 + r4852760;
        double r4852762 = r4852757 - r4852761;
        return r4852762;
}

↓

double f(double x, double y, double z) {
        double r4852763 = x;
        double r4852764 = y;
        double r4852765 = z;
        double r4852766 = fma(r4852763, r4852764, r4852765);
        double r4852767 = r4852766 - r4852765;
        double r4852768 = 1.0;
        double r4852769 = r4852763 * r4852764;
        double r4852770 = r4852768 + r4852769;
        double r4852771 = r4852767 - r4852770;
        double r4852772 = r4852771 * r4852771;
        double r4852773 = r4852772 * r4852771;
        double r4852774 = cbrt(r4852773);
        double r4852775 = cbrt(r4852774);
        double r4852776 = r4852775 * r4852775;
        double r4852777 = r4852776 * r4852775;
        double r4852778 = exp(r4852777);
        double r4852779 = log(r4852778);
        return r4852779;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	45.0
Target	0
Herbie	31.1

\[-1\]

Derivation

1. Initial program 45.0

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

3. Applied add-log-exp 46.6

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

4. Applied add-log-exp 47.6

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

5. Applied sum-log 47.6

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

6. Applied add-log-exp 47.6

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

7. Applied sum-log 47.6

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

8. Applied add-log-exp 47.6

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

9. Applied diff-log 47.6

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

10. Simplified 45.2

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

12. Applied add-log-exp 46.6

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

13. Applied add-log-exp 46.6

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

14. Applied sum-log 46.6

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

15. Applied add-log-exp 47.6

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

16. Applied diff-log 47.6

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

17. Simplified 32.3

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

19. Applied add-cbrt-cube 32.3

    \[\leadsto \log \left(e^{\color{blue}{\sqrt[3]{\left(\left(\log \left(e^{\left(\mathsf{fma}\left(x, y, z\right) - z\right) - 1}\right) - x \cdot y\right) \cdot \left(\log \left(e^{\left(\mathsf{fma}\left(x, y, z\right) - z\right) - 1}\right) - x \cdot y\right)\right) \cdot \left(\log \left(e^{\left(\mathsf{fma}\left(x, y, z\right) - z\right) - 1}\right) - x \cdot y\right)}}}\right)\]

20. Simplified 31.1

    \[\leadsto \log \left(e^{\sqrt[3]{\color{blue}{\left(\left(\mathsf{fma}\left(x, y, z\right) - z\right) - \left(1 + x \cdot y\right)\right) \cdot \left(\left(\left(\mathsf{fma}\left(x, y, z\right) - z\right) - \left(1 + x \cdot y\right)\right) \cdot \left(\left(\mathsf{fma}\left(x, y, z\right) - z\right) - \left(1 + x \cdot y\right)\right)\right)}}}\right)\]
21. Using strategy rm

22. Applied add-cube-cbrt 31.1

    \[\leadsto \log \left(e^{\color{blue}{\left(\sqrt[3]{\sqrt[3]{\left(\left(\mathsf{fma}\left(x, y, z\right) - z\right) - \left(1 + x \cdot y\right)\right) \cdot \left(\left(\left(\mathsf{fma}\left(x, y, z\right) - z\right) - \left(1 + x \cdot y\right)\right) \cdot \left(\left(\mathsf{fma}\left(x, y, z\right) - z\right) - \left(1 + x \cdot y\right)\right)\right)}} \cdot \sqrt[3]{\sqrt[3]{\left(\left(\mathsf{fma}\left(x, y, z\right) - z\right) - \left(1 + x \cdot y\right)\right) \cdot \left(\left(\left(\mathsf{fma}\left(x, y, z\right) - z\right) - \left(1 + x \cdot y\right)\right) \cdot \left(\left(\mathsf{fma}\left(x, y, z\right) - z\right) - \left(1 + x \cdot y\right)\right)\right)}}\right) \cdot \sqrt[3]{\sqrt[3]{\left(\left(\mathsf{fma}\left(x, y, z\right) - z\right) - \left(1 + x \cdot y\right)\right) \cdot \left(\left(\left(\mathsf{fma}\left(x, y, z\right) - z\right) - \left(1 + x \cdot y\right)\right) \cdot \left(\left(\mathsf{fma}\left(x, y, z\right) - z\right) - \left(1 + x \cdot y\right)\right)\right)}}}}\right)\]

23. Final simplification 31.1

    \[\leadsto \log \left(e^{\left(\sqrt[3]{\sqrt[3]{\left(\left(\left(\mathsf{fma}\left(x, y, z\right) - z\right) - \left(1 + x \cdot y\right)\right) \cdot \left(\left(\mathsf{fma}\left(x, y, z\right) - z\right) - \left(1 + x \cdot y\right)\right)\right) \cdot \left(\left(\mathsf{fma}\left(x, y, z\right) - z\right) - \left(1 + x \cdot y\right)\right)}} \cdot \sqrt[3]{\sqrt[3]{\left(\left(\left(\mathsf{fma}\left(x, y, z\right) - z\right) - \left(1 + x \cdot y\right)\right) \cdot \left(\left(\mathsf{fma}\left(x, y, z\right) - z\right) - \left(1 + x \cdot y\right)\right)\right) \cdot \left(\left(\mathsf{fma}\left(x, y, z\right) - z\right) - \left(1 + x \cdot y\right)\right)}}\right) \cdot \sqrt[3]{\sqrt[3]{\left(\left(\left(\mathsf{fma}\left(x, y, z\right) - z\right) - \left(1 + x \cdot y\right)\right) \cdot \left(\left(\mathsf{fma}\left(x, y, z\right) - z\right) - \left(1 + x \cdot y\right)\right)\right) \cdot \left(\left(\mathsf{fma}\left(x, y, z\right) - z\right) - \left(1 + x \cdot y\right)\right)}}}\right)\]

Reproduce

herbie shell --seed 2019172 
(FPCore (x y z)
  :name "simple fma test"

  :herbie-target
  -1.0

  (- (fma x y z) (+ 1.0 (+ (* x y) z))))