Average Error: 0.3 → 0.2
Time: 9.9s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\log \left(x + y\right) + \mathsf{fma}\left(\log t, a - 0.5, \log z - t\right)\]
\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t
↓
\log \left(x + y\right) + \mathsf{fma}\left(\log t, a - 0.5, \log z - t\right)
double f(double x, double y, double z, double t, double a) {
        double r415497 = x;
        double r415498 = y;
        double r415499 = r415497 + r415498;
        double r415500 = log(r415499);
        double r415501 = z;
        double r415502 = log(r415501);
        double r415503 = r415500 + r415502;
        double r415504 = t;
        double r415505 = r415503 - r415504;
        double r415506 = a;
        double r415507 = 0.5;
        double r415508 = r415506 - r415507;
        double r415509 = log(r415504);
        double r415510 = r415508 * r415509;
        double r415511 = r415505 + r415510;
        return r415511;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r415512 = x;
        double r415513 = y;
        double r415514 = r415512 + r415513;
        double r415515 = log(r415514);
        double r415516 = t;
        double r415517 = log(r415516);
        double r415518 = a;
        double r415519 = 0.5;
        double r415520 = r415518 - r415519;
        double r415521 = z;
        double r415522 = log(r415521);
        double r415523 = r415522 - r415516;
        double r415524 = fma(r415517, r415520, r415523);
        double r415525 = r415515 + r415524;
        return r415525;
}

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

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+0.3

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

  5. Simplified0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2020020 +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))))