Average Error: 0.2 → 0.2
Time: 38.0s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\mathsf{fma}\left(\log t, a - 0.5, \log \left(y + x\right) + \left(\log z - t\right)\right)\]
\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, \log \left(y + x\right) + \left(\log z - t\right)\right)
double f(double x, double y, double z, double t, double a) {
        double r78704 = x;
        double r78705 = y;
        double r78706 = r78704 + r78705;
        double r78707 = log(r78706);
        double r78708 = z;
        double r78709 = log(r78708);
        double r78710 = r78707 + r78709;
        double r78711 = t;
        double r78712 = r78710 - r78711;
        double r78713 = a;
        double r78714 = 0.5;
        double r78715 = r78713 - r78714;
        double r78716 = log(r78711);
        double r78717 = r78715 * r78716;
        double r78718 = r78712 + r78717;
        return r78718;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r78719 = t;
        double r78720 = log(r78719);
        double r78721 = a;
        double r78722 = 0.5;
        double r78723 = r78721 - r78722;
        double r78724 = y;
        double r78725 = x;
        double r78726 = r78724 + r78725;
        double r78727 = log(r78726);
        double r78728 = z;
        double r78729 = log(r78728);
        double r78730 = r78729 - r78719;
        double r78731 = r78727 + r78730;
        double r78732 = fma(r78720, r78723, r78731);
        return r78732;
}

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

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

  3. Final simplification0.2

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))