Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Average Error: 0.2 → 0.2

Time: 12.6s

Precision: 64

Math

\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]

↓

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

C

double f(double x, double y, double z, double t, double a) {
        double r319599 = x;
        double r319600 = y;
        double r319601 = r319599 + r319600;
        double r319602 = log(r319601);
        double r319603 = z;
        double r319604 = log(r319603);
        double r319605 = r319602 + r319604;
        double r319606 = t;
        double r319607 = r319605 - r319606;
        double r319608 = a;
        double r319609 = 0.5;
        double r319610 = r319608 - r319609;
        double r319611 = log(r319606);
        double r319612 = r319610 * r319611;
        double r319613 = r319607 + r319612;
        return r319613;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r319614 = t;
        double r319615 = log(r319614);
        double r319616 = a;
        double r319617 = 0.5;
        double r319618 = r319616 - r319617;
        double r319619 = x;
        double r319620 = y;
        double r319621 = r319619 + r319620;
        double r319622 = log(r319621);
        double r319623 = z;
        double r319624 = log(r319623);
        double r319625 = r319622 + r319624;
        double r319626 = r319625 - r319614;
        double r319627 = 1.0;
        double r319628 = pow(r319626, r319627);
        double r319629 = fma(r319615, r319618, r319628);
        return r319629;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	0.2
Target	0.3
Herbie	0.2

\[\log \left(x + y\right) + \left(\left(\log z - t\right) + \left(a - 0.5\right) \cdot \log t\right)\]

Derivation

1. Initial program 0.2

   \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]

2. Simplified 0.2

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

4. Applied pow1 0.2

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

5. Final simplification 0.2

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

Reproduce

herbie shell --seed 2020056 +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))))