Average Error: 0.3 → 0.3
Time: 30.9s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\mathsf{fma}\left(\log t, a - 0.5, \left(\left(\log \left(\sqrt{z}\right) - t\right) + \log \left(\sqrt{z}\right)\right) + \log \left(y + x\right)\right)\]
\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{z}\right) - t\right) + \log \left(\sqrt{z}\right)\right) + \log \left(y + x\right)\right)
double f(double x, double y, double z, double t, double a) {
        double r11816758 = x;
        double r11816759 = y;
        double r11816760 = r11816758 + r11816759;
        double r11816761 = log(r11816760);
        double r11816762 = z;
        double r11816763 = log(r11816762);
        double r11816764 = r11816761 + r11816763;
        double r11816765 = t;
        double r11816766 = r11816764 - r11816765;
        double r11816767 = a;
        double r11816768 = 0.5;
        double r11816769 = r11816767 - r11816768;
        double r11816770 = log(r11816765);
        double r11816771 = r11816769 * r11816770;
        double r11816772 = r11816766 + r11816771;
        return r11816772;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r11816773 = t;
        double r11816774 = log(r11816773);
        double r11816775 = a;
        double r11816776 = 0.5;
        double r11816777 = r11816775 - r11816776;
        double r11816778 = z;
        double r11816779 = sqrt(r11816778);
        double r11816780 = log(r11816779);
        double r11816781 = r11816780 - r11816773;
        double r11816782 = r11816781 + r11816780;
        double r11816783 = y;
        double r11816784 = x;
        double r11816785 = r11816783 + r11816784;
        double r11816786 = log(r11816785);
        double r11816787 = r11816782 + r11816786;
        double r11816788 = fma(r11816774, r11816777, r11816787);
        return r11816788;
}

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-sqr-sqrt0.3

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

  5. Applied log-prod0.3

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

  6. Applied associate--l+0.3

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

  7. Final simplification0.3

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

Reproduce

herbie shell --seed 2019171 +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))))