Average Error: 0.3 → 0.3
Time: 44.5s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\mathsf{fma}\left(a - 0.5, \log t, \left(\log \left(x + y\right) + \log z\right) + \left(-t\right)\right)\]
\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t
↓
\mathsf{fma}\left(a - 0.5, \log t, \left(\log \left(x + y\right) + \log z\right) + \left(-t\right)\right)
double f(double x, double y, double z, double t, double a) {
        double r301345 = x;
        double r301346 = y;
        double r301347 = r301345 + r301346;
        double r301348 = log(r301347);
        double r301349 = z;
        double r301350 = log(r301349);
        double r301351 = r301348 + r301350;
        double r301352 = t;
        double r301353 = r301351 - r301352;
        double r301354 = a;
        double r301355 = 0.5;
        double r301356 = r301354 - r301355;
        double r301357 = log(r301352);
        double r301358 = r301356 * r301357;
        double r301359 = r301353 + r301358;
        return r301359;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r301360 = a;
        double r301361 = 0.5;
        double r301362 = r301360 - r301361;
        double r301363 = t;
        double r301364 = log(r301363);
        double r301365 = x;
        double r301366 = y;
        double r301367 = r301365 + r301366;
        double r301368 = log(r301367);
        double r301369 = z;
        double r301370 = log(r301369);
        double r301371 = r301368 + r301370;
        double r301372 = -r301363;
        double r301373 = r301371 + r301372;
        double r301374 = fma(r301362, r301364, r301373);
        return r301374;
}

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

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

  4. Applied sub-neg0.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2019304 +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))))