Average Error: 0.3 → 0.3
Time: 15.5s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\mathsf{fma}\left(\log t, a - 0.5, \left(\mathsf{fma}\left(2, \log \left(\sqrt[3]{z}\right), \log \left(x + y\right)\right) + \log \left(\sqrt[3]{z}\right)\right) - t\right)\]
\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, \left(\mathsf{fma}\left(2, \log \left(\sqrt[3]{z}\right), \log \left(x + y\right)\right) + \log \left(\sqrt[3]{z}\right)\right) - t\right)
double f(double x, double y, double z, double t, double a) {
        double r63802 = x;
        double r63803 = y;
        double r63804 = r63802 + r63803;
        double r63805 = log(r63804);
        double r63806 = z;
        double r63807 = log(r63806);
        double r63808 = r63805 + r63807;
        double r63809 = t;
        double r63810 = r63808 - r63809;
        double r63811 = a;
        double r63812 = 0.5;
        double r63813 = r63811 - r63812;
        double r63814 = log(r63809);
        double r63815 = r63813 * r63814;
        double r63816 = r63810 + r63815;
        return r63816;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r63817 = t;
        double r63818 = log(r63817);
        double r63819 = a;
        double r63820 = 0.5;
        double r63821 = r63819 - r63820;
        double r63822 = 2.0;
        double r63823 = z;
        double r63824 = cbrt(r63823);
        double r63825 = log(r63824);
        double r63826 = x;
        double r63827 = y;
        double r63828 = r63826 + r63827;
        double r63829 = log(r63828);
        double r63830 = fma(r63822, r63825, r63829);
        double r63831 = r63830 + r63825;
        double r63832 = r63831 - r63817;
        double r63833 = fma(r63818, r63821, r63832);
        return r63833;
}

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. 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 add-cube-cbrt0.3

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

  5. Applied log-prod0.3

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

  6. Applied associate-+r+0.3

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

  7. Simplified0.3

    \[\leadsto \mathsf{fma}\left(\log t, a - 0.5, \left(\color{blue}{\mathsf{fma}\left(2, \log \left(\sqrt[3]{z}\right), \log \left(x + y\right)\right)} + \log \left(\sqrt[3]{z}\right)\right) - t\right)\]

  8. Final simplification0.3

    \[\leadsto \mathsf{fma}\left(\log t, a - 0.5, \left(\mathsf{fma}\left(2, \log \left(\sqrt[3]{z}\right), \log \left(x + y\right)\right) + \log \left(\sqrt[3]{z}\right)\right) - t\right)\]

Reproduce

herbie shell --seed 2020035 +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))))