Average Error: 0.2 → 0.2
Time: 35.7s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\mathsf{fma}\left(a - 0.5, \log t, \left(\log \left(\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}\right) + \left(\log \left(\sqrt[3]{x + y}\right) + \log z\right)\right) - t\right)\]
\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(\log \left(\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}\right) + \left(\log \left(\sqrt[3]{x + y}\right) + \log z\right)\right) - t\right)
double f(double x, double y, double z, double t, double a) {
        double r51987 = x;
        double r51988 = y;
        double r51989 = r51987 + r51988;
        double r51990 = log(r51989);
        double r51991 = z;
        double r51992 = log(r51991);
        double r51993 = r51990 + r51992;
        double r51994 = t;
        double r51995 = r51993 - r51994;
        double r51996 = a;
        double r51997 = 0.5;
        double r51998 = r51996 - r51997;
        double r51999 = log(r51994);
        double r52000 = r51998 * r51999;
        double r52001 = r51995 + r52000;
        return r52001;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r52002 = a;
        double r52003 = 0.5;
        double r52004 = r52002 - r52003;
        double r52005 = t;
        double r52006 = log(r52005);
        double r52007 = x;
        double r52008 = y;
        double r52009 = r52007 + r52008;
        double r52010 = cbrt(r52009);
        double r52011 = r52010 * r52010;
        double r52012 = log(r52011);
        double r52013 = log(r52010);
        double r52014 = z;
        double r52015 = log(r52014);
        double r52016 = r52013 + r52015;
        double r52017 = r52012 + r52016;
        double r52018 = r52017 - r52005;
        double r52019 = fma(r52004, r52006, r52018);
        return r52019;
}

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

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

  4. Applied add-cube-cbrt0.2

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

  5. Applied log-prod0.3

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

  6. Applied associate-+l+0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019235 +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))))