Average Error: 0.3 → 0.3
Time: 38.5s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\left(\log \left(y + x\right) - \mathsf{fma}\left(0.5 - a, \log t, t\right)\right) + \log z\]
\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t
↓
\left(\log \left(y + x\right) - \mathsf{fma}\left(0.5 - a, \log t, t\right)\right) + \log z
double f(double x, double y, double z, double t, double a) {
        double r22262292 = x;
        double r22262293 = y;
        double r22262294 = r22262292 + r22262293;
        double r22262295 = log(r22262294);
        double r22262296 = z;
        double r22262297 = log(r22262296);
        double r22262298 = r22262295 + r22262297;
        double r22262299 = t;
        double r22262300 = r22262298 - r22262299;
        double r22262301 = a;
        double r22262302 = 0.5;
        double r22262303 = r22262301 - r22262302;
        double r22262304 = log(r22262299);
        double r22262305 = r22262303 * r22262304;
        double r22262306 = r22262300 + r22262305;
        return r22262306;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r22262307 = y;
        double r22262308 = x;
        double r22262309 = r22262307 + r22262308;
        double r22262310 = log(r22262309);
        double r22262311 = 0.5;
        double r22262312 = a;
        double r22262313 = r22262311 - r22262312;
        double r22262314 = t;
        double r22262315 = log(r22262314);
        double r22262316 = fma(r22262313, r22262315, r22262314);
        double r22262317 = r22262310 - r22262316;
        double r22262318 = z;
        double r22262319 = log(r22262318);
        double r22262320 = r22262317 + r22262319;
        return r22262320;
}

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

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

  4. Applied sub-neg0.3

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

  5. Applied associate-+l+0.3

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

  6. Simplified0.3

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

  7. Final simplification0.3

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

Reproduce

herbie shell --seed 2019172 +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))))