Average Error: 0.3 → 0.3
Time: 11.2s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\mathsf{fma}\left(\log t, a - 0.5, \frac{{\left(\log \left(x + y\right)\right)}^{3} + {\left(\log z\right)}^{3}}{\mathsf{fma}\left(\log z, \log z - \log \left(x + y\right), \log \left(x + y\right) \cdot \log \left(x + y\right)\right)} - t\right)\]
\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, \frac{{\left(\log \left(x + y\right)\right)}^{3} + {\left(\log z\right)}^{3}}{\mathsf{fma}\left(\log z, \log z - \log \left(x + y\right), \log \left(x + y\right) \cdot \log \left(x + y\right)\right)} - t\right)
double f(double x, double y, double z, double t, double a) {
        double r293592 = x;
        double r293593 = y;
        double r293594 = r293592 + r293593;
        double r293595 = log(r293594);
        double r293596 = z;
        double r293597 = log(r293596);
        double r293598 = r293595 + r293597;
        double r293599 = t;
        double r293600 = r293598 - r293599;
        double r293601 = a;
        double r293602 = 0.5;
        double r293603 = r293601 - r293602;
        double r293604 = log(r293599);
        double r293605 = r293603 * r293604;
        double r293606 = r293600 + r293605;
        return r293606;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r293607 = t;
        double r293608 = log(r293607);
        double r293609 = a;
        double r293610 = 0.5;
        double r293611 = r293609 - r293610;
        double r293612 = x;
        double r293613 = y;
        double r293614 = r293612 + r293613;
        double r293615 = log(r293614);
        double r293616 = 3.0;
        double r293617 = pow(r293615, r293616);
        double r293618 = z;
        double r293619 = log(r293618);
        double r293620 = pow(r293619, r293616);
        double r293621 = r293617 + r293620;
        double r293622 = r293619 - r293615;
        double r293623 = r293615 * r293615;
        double r293624 = fma(r293619, r293622, r293623);
        double r293625 = r293621 / r293624;
        double r293626 = r293625 - r293607;
        double r293627 = fma(r293608, r293611, r293626);
        return r293627;
}

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. Simplified0.3

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

  4. Applied flip3-+0.3

    \[\leadsto \mathsf{fma}\left(\log t, a - 0.5, \color{blue}{\frac{{\left(\log \left(x + y\right)\right)}^{3} + {\left(\log z\right)}^{3}}{\log \left(x + y\right) \cdot \log \left(x + y\right) + \left(\log z \cdot \log z - \log \left(x + y\right) \cdot \log z\right)}} - t\right)\]

  5. Simplified0.3

    \[\leadsto \mathsf{fma}\left(\log t, a - 0.5, \frac{{\left(\log \left(x + y\right)\right)}^{3} + {\left(\log z\right)}^{3}}{\color{blue}{\mathsf{fma}\left(\log z, \log z - \log \left(x + y\right), \log \left(x + y\right) \cdot \log \left(x + y\right)\right)}} - t\right)\]

  6. Final simplification0.3

    \[\leadsto \mathsf{fma}\left(\log t, a - 0.5, \frac{{\left(\log \left(x + y\right)\right)}^{3} + {\left(\log z\right)}^{3}}{\mathsf{fma}\left(\log z, \log z - \log \left(x + y\right), \log \left(x + y\right) \cdot \log \left(x + y\right)\right)} - t\right)\]

Reproduce

herbie shell --seed 2020039 +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))))