Average Error: 0.2 → 0.3
Time: 33.8s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\log \left(\sqrt{x + y}\right) + \left(\left(\log \left(\sqrt{x + y}\right) - t\right) + \mathsf{fma}\left(\left(a - 0.5\right), \left(\log t\right), \left(\log z\right)\right)\right)\]
\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t
↓
\log \left(\sqrt{x + y}\right) + \left(\left(\log \left(\sqrt{x + y}\right) - t\right) + \mathsf{fma}\left(\left(a - 0.5\right), \left(\log t\right), \left(\log z\right)\right)\right)
double f(double x, double y, double z, double t, double a) {
        double r946442 = x;
        double r946443 = y;
        double r946444 = r946442 + r946443;
        double r946445 = log(r946444);
        double r946446 = z;
        double r946447 = log(r946446);
        double r946448 = r946445 + r946447;
        double r946449 = t;
        double r946450 = r946448 - r946449;
        double r946451 = a;
        double r946452 = 0.5;
        double r946453 = r946451 - r946452;
        double r946454 = log(r946449);
        double r946455 = r946453 * r946454;
        double r946456 = r946450 + r946455;
        return r946456;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r946457 = x;
        double r946458 = y;
        double r946459 = r946457 + r946458;
        double r946460 = sqrt(r946459);
        double r946461 = log(r946460);
        double r946462 = t;
        double r946463 = r946461 - r946462;
        double r946464 = a;
        double r946465 = 0.5;
        double r946466 = r946464 - r946465;
        double r946467 = log(r946462);
        double r946468 = z;
        double r946469 = log(r946468);
        double r946470 = fma(r946466, r946467, r946469);
        double r946471 = r946463 + r946470;
        double r946472 = r946461 + r946471;
        return r946472;
}

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

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

  4. Applied add-sqr-sqrt0.2

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

  5. Applied log-prod0.2

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

  6. Applied associate--l+0.3

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

  7. Applied associate-+l+0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2019128 +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))))