Average Error: 0.3 → 0.3
Time: 13.9s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\log \left(x + y\right) + \mathsf{fma}\left(\log t, a - 0.5, \log z - t\right)\]
\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(\log t, a - 0.5, \log z - t\right)
double f(double x, double y, double z, double t, double a) {
        double r338657 = x;
        double r338658 = y;
        double r338659 = r338657 + r338658;
        double r338660 = log(r338659);
        double r338661 = z;
        double r338662 = log(r338661);
        double r338663 = r338660 + r338662;
        double r338664 = t;
        double r338665 = r338663 - r338664;
        double r338666 = a;
        double r338667 = 0.5;
        double r338668 = r338666 - r338667;
        double r338669 = log(r338664);
        double r338670 = r338668 * r338669;
        double r338671 = r338665 + r338670;
        return r338671;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r338672 = x;
        double r338673 = y;
        double r338674 = r338672 + r338673;
        double r338675 = log(r338674);
        double r338676 = t;
        double r338677 = log(r338676);
        double r338678 = a;
        double r338679 = 0.5;
        double r338680 = r338678 - r338679;
        double r338681 = z;
        double r338682 = log(r338681);
        double r338683 = r338682 - r338676;
        double r338684 = fma(r338677, r338680, r338683);
        double r338685 = r338675 + r338684;
        return r338685;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original0.3
Target0.3
Herbie0.3
\[\log \left(x + y\right) + \left(\left(\log z - t\right) + \left(a - 0.5\right) \cdot \log t\right)\]

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

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2020089 +o rules:numerics
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (+ (log (+ x y)) (+ (- (log z) t) (* (- a 0.5) (log t))))

  (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))