Average Error: 0.3 → 0.2
Time: 30.1s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\mathsf{fma}\left(\log t, a - 0.5, \left(\log \left(\sqrt{z}\right) - t\right) + \log \left(\sqrt{z}\right)\right) + \log \left(y + x\right)\]
\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{z}\right) - t\right) + \log \left(\sqrt{z}\right)\right) + \log \left(y + x\right)
double f(double x, double y, double z, double t, double a) {
        double r16110578 = x;
        double r16110579 = y;
        double r16110580 = r16110578 + r16110579;
        double r16110581 = log(r16110580);
        double r16110582 = z;
        double r16110583 = log(r16110582);
        double r16110584 = r16110581 + r16110583;
        double r16110585 = t;
        double r16110586 = r16110584 - r16110585;
        double r16110587 = a;
        double r16110588 = 0.5;
        double r16110589 = r16110587 - r16110588;
        double r16110590 = log(r16110585);
        double r16110591 = r16110589 * r16110590;
        double r16110592 = r16110586 + r16110591;
        return r16110592;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r16110593 = t;
        double r16110594 = log(r16110593);
        double r16110595 = a;
        double r16110596 = 0.5;
        double r16110597 = r16110595 - r16110596;
        double r16110598 = z;
        double r16110599 = sqrt(r16110598);
        double r16110600 = log(r16110599);
        double r16110601 = r16110600 - r16110593;
        double r16110602 = r16110601 + r16110600;
        double r16110603 = fma(r16110594, r16110597, r16110602);
        double r16110604 = y;
        double r16110605 = x;
        double r16110606 = r16110604 + r16110605;
        double r16110607 = log(r16110606);
        double r16110608 = r16110603 + r16110607;
        return r16110608;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original0.3
Target0.2
Herbie0.2
\[\log \left(x + y\right) + \left(\left(\log z - t\right) + \left(a - 0.5\right) \cdot \log t\right)\]

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. Using strategy rm

  3. Applied associate--l+0.3

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

  4. Applied associate-+l+0.2

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

  5. Simplified0.2

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

  7. Applied add-sqr-sqrt0.2

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

  8. Applied log-prod0.2

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

  9. Applied associate--l+0.2

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

  10. Final simplification0.2

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

Reproduce

herbie shell --seed 2019164 +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))))