Average Error: 0.3 → 0.3
Time: 29.3s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\mathsf{fma}\left(\log t, a - 0.5, \left(\left(\log \left(\sqrt[3]{z}\right) - t\right) + \log \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) + \log \left(y + x\right)\right)\]
\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(\left(\log \left(\sqrt[3]{z}\right) - t\right) + \log \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) + \log \left(y + x\right)\right)
double f(double x, double y, double z, double t, double a) {
        double r11992709 = x;
        double r11992710 = y;
        double r11992711 = r11992709 + r11992710;
        double r11992712 = log(r11992711);
        double r11992713 = z;
        double r11992714 = log(r11992713);
        double r11992715 = r11992712 + r11992714;
        double r11992716 = t;
        double r11992717 = r11992715 - r11992716;
        double r11992718 = a;
        double r11992719 = 0.5;
        double r11992720 = r11992718 - r11992719;
        double r11992721 = log(r11992716);
        double r11992722 = r11992720 * r11992721;
        double r11992723 = r11992717 + r11992722;
        return r11992723;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r11992724 = t;
        double r11992725 = log(r11992724);
        double r11992726 = a;
        double r11992727 = 0.5;
        double r11992728 = r11992726 - r11992727;
        double r11992729 = z;
        double r11992730 = cbrt(r11992729);
        double r11992731 = log(r11992730);
        double r11992732 = r11992731 - r11992724;
        double r11992733 = r11992730 * r11992730;
        double r11992734 = log(r11992733);
        double r11992735 = r11992732 + r11992734;
        double r11992736 = y;
        double r11992737 = x;
        double r11992738 = r11992736 + r11992737;
        double r11992739 = log(r11992738);
        double r11992740 = r11992735 + r11992739;
        double r11992741 = fma(r11992725, r11992728, r11992740);
        return r11992741;
}

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

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

  4. Applied add-cube-cbrt0.3

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

  5. Applied log-prod0.3

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

  6. Applied associate--l+0.3

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

  7. Final simplification0.3

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"

  :herbie-target
  (+ (log (+ x y)) (+ (- (log z) t) (* (- a 0.5) (log t))))

  (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))