Average Error: 0.3 → 0.3
Time: 1.0m
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 r1615 = x;
        double r1616 = y;
        double r1617 = r1615 + r1616;
        double r1618 = log(r1617);
        double r1619 = z;
        double r1620 = log(r1619);
        double r1621 = r1618 + r1620;
        double r1622 = t;
        double r1623 = r1621 - r1622;
        double r1624 = a;
        double r1625 = 0.5;
        double r1626 = r1624 - r1625;
        double r1627 = log(r1622);
        double r1628 = r1626 * r1627;
        double r1629 = r1623 + r1628;
        return r1629;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r1630 = t;
        double r1631 = log(r1630);
        double r1632 = a;
        double r1633 = 0.5;
        double r1634 = r1632 - r1633;
        double r1635 = x;
        double r1636 = y;
        double r1637 = r1635 + r1636;
        double r1638 = log(r1637);
        double r1639 = 3.0;
        double r1640 = pow(r1638, r1639);
        double r1641 = z;
        double r1642 = log(r1641);
        double r1643 = pow(r1642, r1639);
        double r1644 = r1640 + r1643;
        double r1645 = r1642 - r1638;
        double r1646 = r1638 * r1638;
        double r1647 = fma(r1642, r1645, r1646);
        double r1648 = r1644 / r1647;
        double r1649 = r1648 - r1630;
        double r1650 = fma(r1631, r1634, r1649);
        return r1650;
}

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 2020025 +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))))