Average Error: 0.3 → 0.3
Time: 12.9s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\mathsf{fma}\left(\log t, a - 0.5, \left(\log \left(\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}\right) + \left(\log \left(\sqrt[3]{x + y}\right) + \log z\right)\right) - t\right)\]
\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(\log \left(\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}\right) + \left(\log \left(\sqrt[3]{x + y}\right) + \log z\right)\right) - t\right)
double f(double x, double y, double z, double t, double a) {
        double r422797 = x;
        double r422798 = y;
        double r422799 = r422797 + r422798;
        double r422800 = log(r422799);
        double r422801 = z;
        double r422802 = log(r422801);
        double r422803 = r422800 + r422802;
        double r422804 = t;
        double r422805 = r422803 - r422804;
        double r422806 = a;
        double r422807 = 0.5;
        double r422808 = r422806 - r422807;
        double r422809 = log(r422804);
        double r422810 = r422808 * r422809;
        double r422811 = r422805 + r422810;
        return r422811;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r422812 = t;
        double r422813 = log(r422812);
        double r422814 = a;
        double r422815 = 0.5;
        double r422816 = r422814 - r422815;
        double r422817 = x;
        double r422818 = y;
        double r422819 = r422817 + r422818;
        double r422820 = cbrt(r422819);
        double r422821 = r422820 * r422820;
        double r422822 = log(r422821);
        double r422823 = log(r422820);
        double r422824 = z;
        double r422825 = log(r422824);
        double r422826 = r422823 + r422825;
        double r422827 = r422822 + r422826;
        double r422828 = r422827 - r422812;
        double r422829 = fma(r422813, r422816, r422828);
        return r422829;
}

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

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

  4. Applied add-cube-cbrt0.3

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

  5. Applied log-prod0.3

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

  6. Applied associate-+l+0.3

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

  7. Final simplification0.3

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

Reproduce

herbie shell --seed 2019362 +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))))