Average Error: 0.3 → 0.3
Time: 12.2s
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 r295281 = x;
        double r295282 = y;
        double r295283 = r295281 + r295282;
        double r295284 = log(r295283);
        double r295285 = z;
        double r295286 = log(r295285);
        double r295287 = r295284 + r295286;
        double r295288 = t;
        double r295289 = r295287 - r295288;
        double r295290 = a;
        double r295291 = 0.5;
        double r295292 = r295290 - r295291;
        double r295293 = log(r295288);
        double r295294 = r295292 * r295293;
        double r295295 = r295289 + r295294;
        return r295295;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r295296 = x;
        double r295297 = y;
        double r295298 = r295296 + r295297;
        double r295299 = log(r295298);
        double r295300 = t;
        double r295301 = log(r295300);
        double r295302 = a;
        double r295303 = 0.5;
        double r295304 = r295302 - r295303;
        double r295305 = z;
        double r295306 = log(r295305);
        double r295307 = r295306 - r295300;
        double r295308 = fma(r295301, r295304, r295307);
        double r295309 = r295299 + r295308;
        return r295309;
}

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. 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. Simplified0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2020021 +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))))