Average Error: 0.2 → 0.4
Time: 15.9s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\mathsf{fma}\left(\log t, a - 0.5, \mathsf{fma}\left(\sqrt[3]{\log \left(x + y\right)} \cdot \sqrt[3]{\log \left(x + y\right)}, \sqrt[3]{\log \left(x + y\right)}, \log z\right) - t\right)\]
\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t
↓
\mathsf{fma}\left(\log t, a - 0.5, \mathsf{fma}\left(\sqrt[3]{\log \left(x + y\right)} \cdot \sqrt[3]{\log \left(x + y\right)}, \sqrt[3]{\log \left(x + y\right)}, \log z\right) - t\right)
double f(double x, double y, double z, double t, double a) {
        double r80698 = x;
        double r80699 = y;
        double r80700 = r80698 + r80699;
        double r80701 = log(r80700);
        double r80702 = z;
        double r80703 = log(r80702);
        double r80704 = r80701 + r80703;
        double r80705 = t;
        double r80706 = r80704 - r80705;
        double r80707 = a;
        double r80708 = 0.5;
        double r80709 = r80707 - r80708;
        double r80710 = log(r80705);
        double r80711 = r80709 * r80710;
        double r80712 = r80706 + r80711;
        return r80712;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r80713 = t;
        double r80714 = log(r80713);
        double r80715 = a;
        double r80716 = 0.5;
        double r80717 = r80715 - r80716;
        double r80718 = x;
        double r80719 = y;
        double r80720 = r80718 + r80719;
        double r80721 = log(r80720);
        double r80722 = cbrt(r80721);
        double r80723 = r80722 * r80722;
        double r80724 = z;
        double r80725 = log(r80724);
        double r80726 = fma(r80723, r80722, r80725);
        double r80727 = r80726 - r80713;
        double r80728 = fma(r80714, r80717, r80727);
        return r80728;
}

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 0.2

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

  2. Simplified0.2

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

  4. Applied add-cube-cbrt0.4

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

  5. Applied fma-def0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2020083 +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))))