Average Error: 0.3 → 0.3
Time: 11.7s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\mathsf{fma}\left(\log t, a - 0.5, \left(\log \left(x + y\right) + \log z\right) - t\right)\]
\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, \left(\log \left(x + y\right) + \log z\right) - t\right)
double f(double x, double y, double z, double t, double a) {
        double r81052 = x;
        double r81053 = y;
        double r81054 = r81052 + r81053;
        double r81055 = log(r81054);
        double r81056 = z;
        double r81057 = log(r81056);
        double r81058 = r81055 + r81057;
        double r81059 = t;
        double r81060 = r81058 - r81059;
        double r81061 = a;
        double r81062 = 0.5;
        double r81063 = r81061 - r81062;
        double r81064 = log(r81059);
        double r81065 = r81063 * r81064;
        double r81066 = r81060 + r81065;
        return r81066;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r81067 = t;
        double r81068 = log(r81067);
        double r81069 = a;
        double r81070 = 0.5;
        double r81071 = r81069 - r81070;
        double r81072 = x;
        double r81073 = y;
        double r81074 = r81072 + r81073;
        double r81075 = log(r81074);
        double r81076 = z;
        double r81077 = log(r81076);
        double r81078 = r81075 + r81077;
        double r81079 = r81078 - r81067;
        double r81080 = fma(r81068, r81071, r81079);
        return r81080;
}

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

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

  4. Applied *-un-lft-identity0.3

    \[\leadsto \color{blue}{1 \cdot \mathsf{fma}\left(\log t, a - 0.5, \left(\log \left(x + y\right) + \log z\right) - t\right)}\]

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2019354 +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))))