Average Error: 0.3 → 0.3
Time: 11.4s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\mathsf{fma}\left(\log t, a - 0.5, \frac{\log \left(x + y\right) \cdot \log \left(\sqrt{x + y}\right) + \mathsf{fma}\left(\log \left(\sqrt{x + y}\right), \log \left(x + y\right), -{\left(\log z\right)}^{2}\right)}{\log \left(x + y\right) - \log z} - t\right)\]
\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{\log \left(x + y\right) \cdot \log \left(\sqrt{x + y}\right) + \mathsf{fma}\left(\log \left(\sqrt{x + y}\right), \log \left(x + y\right), -{\left(\log z\right)}^{2}\right)}{\log \left(x + y\right) - \log z} - t\right)
double f(double x, double y, double z, double t, double a) {
        double r58847 = x;
        double r58848 = y;
        double r58849 = r58847 + r58848;
        double r58850 = log(r58849);
        double r58851 = z;
        double r58852 = log(r58851);
        double r58853 = r58850 + r58852;
        double r58854 = t;
        double r58855 = r58853 - r58854;
        double r58856 = a;
        double r58857 = 0.5;
        double r58858 = r58856 - r58857;
        double r58859 = log(r58854);
        double r58860 = r58858 * r58859;
        double r58861 = r58855 + r58860;
        return r58861;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r58862 = t;
        double r58863 = log(r58862);
        double r58864 = a;
        double r58865 = 0.5;
        double r58866 = r58864 - r58865;
        double r58867 = x;
        double r58868 = y;
        double r58869 = r58867 + r58868;
        double r58870 = log(r58869);
        double r58871 = sqrt(r58869);
        double r58872 = log(r58871);
        double r58873 = r58870 * r58872;
        double r58874 = z;
        double r58875 = log(r58874);
        double r58876 = 2.0;
        double r58877 = pow(r58875, r58876);
        double r58878 = -r58877;
        double r58879 = fma(r58872, r58870, r58878);
        double r58880 = r58873 + r58879;
        double r58881 = r58870 - r58875;
        double r58882 = r58880 / r58881;
        double r58883 = r58882 - r58862;
        double r58884 = fma(r58863, r58866, r58883);
        return r58884;
}

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 flip-+0.3

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

  6. Applied add-sqr-sqrt0.3

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

  7. Applied log-prod0.3

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

  8. Applied distribute-lft-in0.3

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

  9. Applied associate--l+0.3

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

  10. Simplified0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2020062 +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))))