Average Error: 0.3 → 0.3
Time: 1.3m
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\mathsf{fma}\left(a - 0.5, \log t, \left(\mathsf{fma}\left(\log \left(\sqrt[3]{\sqrt{z}}\right), 2, \log \left(\sqrt{z}\right) + \log \left(x + y\right)\right) + \log \left(\sqrt[3]{\sqrt{z}}\right)\right) - t\right)\]
\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t
↓
\mathsf{fma}\left(a - 0.5, \log t, \left(\mathsf{fma}\left(\log \left(\sqrt[3]{\sqrt{z}}\right), 2, \log \left(\sqrt{z}\right) + \log \left(x + y\right)\right) + \log \left(\sqrt[3]{\sqrt{z}}\right)\right) - t\right)
double f(double x, double y, double z, double t, double a) {
        double r49979 = x;
        double r49980 = y;
        double r49981 = r49979 + r49980;
        double r49982 = log(r49981);
        double r49983 = z;
        double r49984 = log(r49983);
        double r49985 = r49982 + r49984;
        double r49986 = t;
        double r49987 = r49985 - r49986;
        double r49988 = a;
        double r49989 = 0.5;
        double r49990 = r49988 - r49989;
        double r49991 = log(r49986);
        double r49992 = r49990 * r49991;
        double r49993 = r49987 + r49992;
        return r49993;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r49994 = a;
        double r49995 = 0.5;
        double r49996 = r49994 - r49995;
        double r49997 = t;
        double r49998 = log(r49997);
        double r49999 = z;
        double r50000 = sqrt(r49999);
        double r50001 = cbrt(r50000);
        double r50002 = log(r50001);
        double r50003 = 2.0;
        double r50004 = log(r50000);
        double r50005 = x;
        double r50006 = y;
        double r50007 = r50005 + r50006;
        double r50008 = log(r50007);
        double r50009 = r50004 + r50008;
        double r50010 = fma(r50002, r50003, r50009);
        double r50011 = r50010 + r50002;
        double r50012 = r50011 - r49997;
        double r50013 = fma(r49996, r49998, r50012);
        return r50013;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Derivation

  1. Initial program 0.3

    \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]

  2. Simplified0.3

    \[\leadsto \color{blue}{\mathsf{fma}\left(a - 0.5, \log t, \left(\log \left(x + y\right) + \log z\right) - t\right)}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt0.3

    \[\leadsto \mathsf{fma}\left(a - 0.5, \log t, \left(\log \left(x + y\right) + \log \color{blue}{\left(\sqrt{z} \cdot \sqrt{z}\right)}\right) - t\right)\]

  5. Applied log-prod0.3

    \[\leadsto \mathsf{fma}\left(a - 0.5, \log t, \left(\log \left(x + y\right) + \color{blue}{\left(\log \left(\sqrt{z}\right) + \log \left(\sqrt{z}\right)\right)}\right) - t\right)\]

  6. Applied associate-+r+0.3

    \[\leadsto \mathsf{fma}\left(a - 0.5, \log t, \color{blue}{\left(\left(\log \left(x + y\right) + \log \left(\sqrt{z}\right)\right) + \log \left(\sqrt{z}\right)\right)} - t\right)\]
  7. Using strategy rm

  8. Applied add-cube-cbrt0.3

    \[\leadsto \mathsf{fma}\left(a - 0.5, \log t, \left(\left(\log \left(x + y\right) + \log \left(\sqrt{z}\right)\right) + \log \color{blue}{\left(\left(\sqrt[3]{\sqrt{z}} \cdot \sqrt[3]{\sqrt{z}}\right) \cdot \sqrt[3]{\sqrt{z}}\right)}\right) - t\right)\]

  9. Applied log-prod0.3

    \[\leadsto \mathsf{fma}\left(a - 0.5, \log t, \left(\left(\log \left(x + y\right) + \log \left(\sqrt{z}\right)\right) + \color{blue}{\left(\log \left(\sqrt[3]{\sqrt{z}} \cdot \sqrt[3]{\sqrt{z}}\right) + \log \left(\sqrt[3]{\sqrt{z}}\right)\right)}\right) - t\right)\]

  10. Applied associate-+r+0.3

    \[\leadsto \mathsf{fma}\left(a - 0.5, \log t, \color{blue}{\left(\left(\left(\log \left(x + y\right) + \log \left(\sqrt{z}\right)\right) + \log \left(\sqrt[3]{\sqrt{z}} \cdot \sqrt[3]{\sqrt{z}}\right)\right) + \log \left(\sqrt[3]{\sqrt{z}}\right)\right)} - t\right)\]

  11. Simplified0.3

    \[\leadsto \mathsf{fma}\left(a - 0.5, \log t, \left(\color{blue}{\mathsf{fma}\left(\log \left(\sqrt[3]{\sqrt{z}}\right), 2, \log \left(\sqrt{z}\right) + \log \left(x + y\right)\right)} + \log \left(\sqrt[3]{\sqrt{z}}\right)\right) - t\right)\]

  12. Final simplification0.3

    \[\leadsto \mathsf{fma}\left(a - 0.5, \log t, \left(\mathsf{fma}\left(\log \left(\sqrt[3]{\sqrt{z}}\right), 2, \log \left(\sqrt{z}\right) + \log \left(x + y\right)\right) + \log \left(\sqrt[3]{\sqrt{z}}\right)\right) - t\right)\]

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  :precision binary64
  (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))