Average Error: 0.3 → 0.3
Time: 22.6s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\mathsf{fma}\left(a - 0.5, \log \left(\sqrt{t}\right), \log \left(x + y\right) + \left(\log z - t\right)\right) + \log \left(\sqrt{t}\right) \cdot \left(a - 0.5\right)\]
\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 \left(\sqrt{t}\right), \log \left(x + y\right) + \left(\log z - t\right)\right) + \log \left(\sqrt{t}\right) \cdot \left(a - 0.5\right)
double f(double x, double y, double z, double t, double a) {
        double r72410 = x;
        double r72411 = y;
        double r72412 = r72410 + r72411;
        double r72413 = log(r72412);
        double r72414 = z;
        double r72415 = log(r72414);
        double r72416 = r72413 + r72415;
        double r72417 = t;
        double r72418 = r72416 - r72417;
        double r72419 = a;
        double r72420 = 0.5;
        double r72421 = r72419 - r72420;
        double r72422 = log(r72417);
        double r72423 = r72421 * r72422;
        double r72424 = r72418 + r72423;
        return r72424;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r72425 = a;
        double r72426 = 0.5;
        double r72427 = r72425 - r72426;
        double r72428 = t;
        double r72429 = sqrt(r72428);
        double r72430 = log(r72429);
        double r72431 = x;
        double r72432 = y;
        double r72433 = r72431 + r72432;
        double r72434 = log(r72433);
        double r72435 = z;
        double r72436 = log(r72435);
        double r72437 = r72436 - r72428;
        double r72438 = r72434 + r72437;
        double r72439 = fma(r72427, r72430, r72438);
        double r72440 = r72430 * r72427;
        double r72441 = r72439 + r72440;
        return r72441;
}

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 add-sqr-sqrt0.3

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

  4. Applied log-prod0.3

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

  5. Applied distribute-rgt-in0.3

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

  6. Applied associate-+r+0.3

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

  7. Simplified0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2020043 +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))))