simple fma test

Average Error: 45.1 → 20.1

Time: 21.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -27766186.0693160183727741241455078125:\\ \;\;\;\;\left(\sqrt[3]{\left(\left(\mathsf{fma}\left(x, y, z\right) - z\right) - 1\right) - x \cdot y} \cdot \sqrt[3]{\left(\left(\mathsf{fma}\left(x, y, z\right) - z\right) - 1\right) - x \cdot y}\right) \cdot \sqrt[3]{\sqrt[3]{\left(\left(\mathsf{fma}\left(x, y, z\right) - z\right) - 1\right) - x \cdot y} \cdot \left(\sqrt[3]{\left(\left(\mathsf{fma}\left(x, y, z\right) - z\right) - 1\right) - x \cdot y} \cdot \sqrt[3]{\left(\left(\mathsf{fma}\left(x, y, z\right) - z\right) - 1\right) - x \cdot y}\right)}\\ \mathbf{elif}\;z \le 242525515658638.96875:\\ \;\;\;\;\log \left(e^{\left(\mathsf{fma}\left(x, y, z\right) - x \cdot y\right) - \left(z + 1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\left(\mathsf{fma}\left(x, y, z\right) - z\right) - 1\right) - x \cdot y} \cdot \left(\sqrt[3]{\left(\left(\mathsf{fma}\left(x, y, z\right) - z\right) - 1\right) - x \cdot y} \cdot \sqrt[3]{\left(\left(\mathsf{fma}\left(x, y, z\right) - z\right) - 1\right) - x \cdot y}\right)\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r4421900 = x;
        double r4421901 = y;
        double r4421902 = z;
        double r4421903 = fma(r4421900, r4421901, r4421902);
        double r4421904 = 1.0;
        double r4421905 = r4421900 * r4421901;
        double r4421906 = r4421905 + r4421902;
        double r4421907 = r4421904 + r4421906;
        double r4421908 = r4421903 - r4421907;
        return r4421908;
}

↓

double f(double x, double y, double z) {
        double r4421909 = z;
        double r4421910 = -27766186.06931602;
        bool r4421911 = r4421909 <= r4421910;
        double r4421912 = x;
        double r4421913 = y;
        double r4421914 = fma(r4421912, r4421913, r4421909);
        double r4421915 = r4421914 - r4421909;
        double r4421916 = 1.0;
        double r4421917 = r4421915 - r4421916;
        double r4421918 = r4421912 * r4421913;
        double r4421919 = r4421917 - r4421918;
        double r4421920 = cbrt(r4421919);
        double r4421921 = r4421920 * r4421920;
        double r4421922 = r4421920 * r4421921;
        double r4421923 = cbrt(r4421922);
        double r4421924 = r4421921 * r4421923;
        double r4421925 = 242525515658638.97;
        bool r4421926 = r4421909 <= r4421925;
        double r4421927 = r4421914 - r4421918;
        double r4421928 = r4421909 + r4421916;
        double r4421929 = r4421927 - r4421928;
        double r4421930 = exp(r4421929);
        double r4421931 = log(r4421930);
        double r4421932 = r4421926 ? r4421931 : r4421922;
        double r4421933 = r4421911 ? r4421924 : r4421932;
        return r4421933;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	45.1
Target	0
Herbie	20.1

\[-1\]

Derivation

1. Split input into 3 regimes

2. if z < -27766186.06931602

   1. Initial program 61.4

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

   3. Applied add-log-exp 64.0

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

   4. Applied add-log-exp 64.0

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

   5. Applied sum-log 64.0

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

   6. Applied add-log-exp 64.0

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

   7. Applied sum-log 64.0

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

   8. Applied add-log-exp 64.0

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

   9. Applied diff-log 64.0

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

   10. Simplified 33.8

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

   12. Applied exp-diff 34.2

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

   13. Applied log-div 34.2

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

   14. Simplified 34.2

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

   15. Simplified 33.3

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

   17. Applied add-cube-cbrt 33.3

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

   19. Applied add-cube-cbrt 33.3

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

   if -27766186.06931602 < z < 242525515658638.97

   1. Initial program 29.7

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

   3. Applied add-log-exp 30.7

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

   4. Applied add-log-exp 32.8

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

   5. Applied sum-log 32.8

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

   6. Applied add-log-exp 32.8

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

   7. Applied sum-log 32.8

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

   8. Applied add-log-exp 32.8

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

   9. Applied diff-log 32.8

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

   10. Simplified 29.8

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

   12. Applied exp-diff 31.9

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

   13. Applied log-div 31.9

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

   14. Simplified 31.9

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

   15. Simplified 29.7

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

   17. Applied add-log-exp 31.9

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

   18. Applied add-log-exp 31.9

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

   19. Applied add-log-exp 32.8

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

   20. Applied add-log-exp 32.8

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

   21. Applied diff-log 32.8

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

   22. Applied diff-log 32.8

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

   23. Applied diff-log 32.8

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

   24. Simplified 9.0

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

   if 242525515658638.97 < z

   1. Initial program 62.1

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

   3. Applied add-log-exp 64.0

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

   4. Applied add-log-exp 64.0

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

   5. Applied sum-log 64.0

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

   6. Applied add-log-exp 64.0

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

   7. Applied sum-log 64.0

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

   8. Applied add-log-exp 64.0

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

   9. Applied diff-log 64.0

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

   10. Simplified 31.7

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

   12. Applied exp-diff 32.0

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

   13. Applied log-div 32.0

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

   14. Simplified 32.0

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

   15. Simplified 31.2

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

   17. Applied add-cube-cbrt 31.2

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

4. Final simplification 20.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -27766186.0693160183727741241455078125:\\ \;\;\;\;\left(\sqrt[3]{\left(\left(\mathsf{fma}\left(x, y, z\right) - z\right) - 1\right) - x \cdot y} \cdot \sqrt[3]{\left(\left(\mathsf{fma}\left(x, y, z\right) - z\right) - 1\right) - x \cdot y}\right) \cdot \sqrt[3]{\sqrt[3]{\left(\left(\mathsf{fma}\left(x, y, z\right) - z\right) - 1\right) - x \cdot y} \cdot \left(\sqrt[3]{\left(\left(\mathsf{fma}\left(x, y, z\right) - z\right) - 1\right) - x \cdot y} \cdot \sqrt[3]{\left(\left(\mathsf{fma}\left(x, y, z\right) - z\right) - 1\right) - x \cdot y}\right)}\\ \mathbf{elif}\;z \le 242525515658638.96875:\\ \;\;\;\;\log \left(e^{\left(\mathsf{fma}\left(x, y, z\right) - x \cdot y\right) - \left(z + 1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\left(\mathsf{fma}\left(x, y, z\right) - z\right) - 1\right) - x \cdot y} \cdot \left(\sqrt[3]{\left(\left(\mathsf{fma}\left(x, y, z\right) - z\right) - 1\right) - x \cdot y} \cdot \sqrt[3]{\left(\left(\mathsf{fma}\left(x, y, z\right) - z\right) - 1\right) - x \cdot y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019169 
(FPCore (x y z)
  :name "simple fma test"

  :herbie-target
  -1.0

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