Average Error: 0.2 → 0.3
Time: 13.0s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\mathsf{fma}\left(\log t, a - 0.5, \left(\left(\log \left(x + y\right) + \log \left(\sqrt{z}\right)\right) + \log \left(\sqrt{z}\right)\right) - t\right)\]
\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(\left(\log \left(x + y\right) + \log \left(\sqrt{z}\right)\right) + \log \left(\sqrt{z}\right)\right) - t\right)
double f(double x, double y, double z, double t, double a) {
        double r460698 = x;
        double r460699 = y;
        double r460700 = r460698 + r460699;
        double r460701 = log(r460700);
        double r460702 = z;
        double r460703 = log(r460702);
        double r460704 = r460701 + r460703;
        double r460705 = t;
        double r460706 = r460704 - r460705;
        double r460707 = a;
        double r460708 = 0.5;
        double r460709 = r460707 - r460708;
        double r460710 = log(r460705);
        double r460711 = r460709 * r460710;
        double r460712 = r460706 + r460711;
        return r460712;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r460713 = t;
        double r460714 = log(r460713);
        double r460715 = a;
        double r460716 = 0.5;
        double r460717 = r460715 - r460716;
        double r460718 = x;
        double r460719 = y;
        double r460720 = r460718 + r460719;
        double r460721 = log(r460720);
        double r460722 = z;
        double r460723 = sqrt(r460722);
        double r460724 = log(r460723);
        double r460725 = r460721 + r460724;
        double r460726 = r460725 + r460724;
        double r460727 = r460726 - r460713;
        double r460728 = fma(r460714, r460717, r460727);
        return r460728;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original0.2
Target0.2
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 0.2

    \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]

  2. Simplified0.2

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

  4. Applied add-sqr-sqrt0.2

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

  5. Applied log-prod0.2

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

  6. Applied associate-+r+0.3

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

  7. Final simplification0.3

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

Reproduce

herbie shell --seed 2020100 +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))))