Average Error: 0.3 → 0.3
Time: 39.6s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\mathsf{fma}\left(a - 0.5, \log t, \left(2 \cdot \log \left(\sqrt[3]{z}\right) + \left(\log \left(\sqrt[3]{z}\right) + \log \left(x + y\right)\right)\right) - t\right)\]
\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(2 \cdot \log \left(\sqrt[3]{z}\right) + \left(\log \left(\sqrt[3]{z}\right) + \log \left(x + y\right)\right)\right) - t\right)
double f(double x, double y, double z, double t, double a) {
        double r193947 = x;
        double r193948 = y;
        double r193949 = r193947 + r193948;
        double r193950 = log(r193949);
        double r193951 = z;
        double r193952 = log(r193951);
        double r193953 = r193950 + r193952;
        double r193954 = t;
        double r193955 = r193953 - r193954;
        double r193956 = a;
        double r193957 = 0.5;
        double r193958 = r193956 - r193957;
        double r193959 = log(r193954);
        double r193960 = r193958 * r193959;
        double r193961 = r193955 + r193960;
        return r193961;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r193962 = a;
        double r193963 = 0.5;
        double r193964 = r193962 - r193963;
        double r193965 = t;
        double r193966 = log(r193965);
        double r193967 = 2.0;
        double r193968 = z;
        double r193969 = cbrt(r193968);
        double r193970 = log(r193969);
        double r193971 = r193967 * r193970;
        double r193972 = x;
        double r193973 = y;
        double r193974 = r193972 + r193973;
        double r193975 = log(r193974);
        double r193976 = r193970 + r193975;
        double r193977 = r193971 + r193976;
        double r193978 = r193977 - r193965;
        double r193979 = fma(r193964, r193966, r193978);
        return r193979;
}

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

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

  4. Applied add-cube-cbrt0.3

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

  5. Applied log-prod0.3

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

  6. Applied associate-+r+0.3

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

  7. Simplified0.3

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

  9. Applied fma-udef0.3

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

  10. Applied associate-+l+0.3

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

  11. Simplified0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2019323 +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))))