Average Error: 0.3 → 0.3
Time: 30.9s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\log \left(x + y\right) + \mathsf{fma}\left(a - 0.5, \log t, \log z - t\right)\]
\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t
↓
\log \left(x + y\right) + \mathsf{fma}\left(a - 0.5, \log t, \log z - t\right)
double f(double x, double y, double z, double t, double a) {
        double r52950 = x;
        double r52951 = y;
        double r52952 = r52950 + r52951;
        double r52953 = log(r52952);
        double r52954 = z;
        double r52955 = log(r52954);
        double r52956 = r52953 + r52955;
        double r52957 = t;
        double r52958 = r52956 - r52957;
        double r52959 = a;
        double r52960 = 0.5;
        double r52961 = r52959 - r52960;
        double r52962 = log(r52957);
        double r52963 = r52961 * r52962;
        double r52964 = r52958 + r52963;
        return r52964;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r52965 = x;
        double r52966 = y;
        double r52967 = r52965 + r52966;
        double r52968 = log(r52967);
        double r52969 = a;
        double r52970 = 0.5;
        double r52971 = r52969 - r52970;
        double r52972 = t;
        double r52973 = log(r52972);
        double r52974 = z;
        double r52975 = log(r52974);
        double r52976 = r52975 - r52972;
        double r52977 = fma(r52971, r52973, r52976);
        double r52978 = r52968 + r52977;
        return r52978;
}

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. Using strategy rm

  3. Applied associate--l+0.3

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

  4. Applied associate-+l+0.3

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

  5. Simplified0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019212 +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))))