Average Error: 0.2 → 0.3
Time: 43.3s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\mathsf{fma}\left(a - 0.5, \log t, \left(\left(\log \left(x + y\right) + \log \left(\sqrt{z}\right)\right) + \log \left(\sqrt{z}\right)\right) - t\right)\]
\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t
↓
\mathsf{fma}\left(a - 0.5, \log t, \left(\left(\log \left(x + y\right) + \log \left(\sqrt{z}\right)\right) + \log \left(\sqrt{z}\right)\right) - t\right)
double f(double x, double y, double z, double t, double a) {
        double r238831 = x;
        double r238832 = y;
        double r238833 = r238831 + r238832;
        double r238834 = log(r238833);
        double r238835 = z;
        double r238836 = log(r238835);
        double r238837 = r238834 + r238836;
        double r238838 = t;
        double r238839 = r238837 - r238838;
        double r238840 = a;
        double r238841 = 0.5;
        double r238842 = r238840 - r238841;
        double r238843 = log(r238838);
        double r238844 = r238842 * r238843;
        double r238845 = r238839 + r238844;
        return r238845;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r238846 = a;
        double r238847 = 0.5;
        double r238848 = r238846 - r238847;
        double r238849 = t;
        double r238850 = log(r238849);
        double r238851 = x;
        double r238852 = y;
        double r238853 = r238851 + r238852;
        double r238854 = log(r238853);
        double r238855 = z;
        double r238856 = sqrt(r238855);
        double r238857 = log(r238856);
        double r238858 = r238854 + r238857;
        double r238859 = r238858 + r238857;
        double r238860 = r238859 - r238849;
        double r238861 = fma(r238848, r238850, r238860);
        return r238861;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original0.2
Target0.2
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 0.2

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

  2. Simplified0.2

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

  4. Applied add-sqr-sqrt0.2

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

  5. Applied log-prod0.2

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

  6. Applied associate-+r+0.3

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

  7. Final simplification0.3

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

Reproduce

herbie shell --seed 2019347 +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))))