Average Error: 0.3 → 0.2
Time: 36.2s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\mathsf{fma}\left(\log t, a - 0.5, \left(\log \left(\sqrt{z}\right) - t\right) + \log \left(\sqrt{z}\right)\right) + \log \left(y + x\right)\]
\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(\log \left(\sqrt{z}\right) - t\right) + \log \left(\sqrt{z}\right)\right) + \log \left(y + x\right)
double f(double x, double y, double z, double t, double a) {
        double r2637927 = x;
        double r2637928 = y;
        double r2637929 = r2637927 + r2637928;
        double r2637930 = log(r2637929);
        double r2637931 = z;
        double r2637932 = log(r2637931);
        double r2637933 = r2637930 + r2637932;
        double r2637934 = t;
        double r2637935 = r2637933 - r2637934;
        double r2637936 = a;
        double r2637937 = 0.5;
        double r2637938 = r2637936 - r2637937;
        double r2637939 = log(r2637934);
        double r2637940 = r2637938 * r2637939;
        double r2637941 = r2637935 + r2637940;
        return r2637941;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r2637942 = t;
        double r2637943 = log(r2637942);
        double r2637944 = a;
        double r2637945 = 0.5;
        double r2637946 = r2637944 - r2637945;
        double r2637947 = z;
        double r2637948 = sqrt(r2637947);
        double r2637949 = log(r2637948);
        double r2637950 = r2637949 - r2637942;
        double r2637951 = r2637950 + r2637949;
        double r2637952 = fma(r2637943, r2637946, r2637951);
        double r2637953 = y;
        double r2637954 = x;
        double r2637955 = r2637953 + r2637954;
        double r2637956 = log(r2637955);
        double r2637957 = r2637952 + r2637956;
        return r2637957;
}

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. Using strategy rm

  3. Applied associate--l+0.3

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

  4. Applied associate-+l+0.2

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

  5. Simplified0.2

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

  7. Applied add-sqr-sqrt0.2

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

  8. Applied log-prod0.2

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

  9. Applied associate--l+0.2

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

  10. Final simplification0.2

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

Reproduce

herbie shell --seed 2019164 +o rules:numerics
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))