Average Error: 0.3 → 0.3
Time: 41.7s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\mathsf{fma}\left(a - 0.5, \log t, \left(2 \cdot \log \left(\sqrt[3]{z}\right) + \left(\log \left(\sqrt[3]{z}\right) + \log \left(x + y\right)\right)\right) - t\right)\]
\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(2 \cdot \log \left(\sqrt[3]{z}\right) + \left(\log \left(\sqrt[3]{z}\right) + \log \left(x + y\right)\right)\right) - t\right)
double f(double x, double y, double z, double t, double a) {
        double r284286 = x;
        double r284287 = y;
        double r284288 = r284286 + r284287;
        double r284289 = log(r284288);
        double r284290 = z;
        double r284291 = log(r284290);
        double r284292 = r284289 + r284291;
        double r284293 = t;
        double r284294 = r284292 - r284293;
        double r284295 = a;
        double r284296 = 0.5;
        double r284297 = r284295 - r284296;
        double r284298 = log(r284293);
        double r284299 = r284297 * r284298;
        double r284300 = r284294 + r284299;
        return r284300;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r284301 = a;
        double r284302 = 0.5;
        double r284303 = r284301 - r284302;
        double r284304 = t;
        double r284305 = log(r284304);
        double r284306 = 2.0;
        double r284307 = z;
        double r284308 = cbrt(r284307);
        double r284309 = log(r284308);
        double r284310 = r284306 * r284309;
        double r284311 = x;
        double r284312 = y;
        double r284313 = r284311 + r284312;
        double r284314 = log(r284313);
        double r284315 = r284309 + r284314;
        double r284316 = r284310 + r284315;
        double r284317 = r284316 - r284304;
        double r284318 = fma(r284303, r284305, r284317);
        return r284318;
}

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 add-cube-cbrt0.3

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

  5. Applied log-prod0.3

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

  6. Applied associate-+r+0.3

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

  7. Simplified0.3

    \[\leadsto \mathsf{fma}\left(a - 0.5, \log t, \left(\color{blue}{\mathsf{fma}\left(2, \log \left(\sqrt[3]{z}\right), \log \left(x + y\right)\right)} + \log \left(\sqrt[3]{z}\right)\right) - t\right)\]
  8. Using strategy rm

  9. Applied fma-udef0.3

    \[\leadsto \mathsf{fma}\left(a - 0.5, \log t, \left(\color{blue}{\left(2 \cdot \log \left(\sqrt[3]{z}\right) + \log \left(x + y\right)\right)} + \log \left(\sqrt[3]{z}\right)\right) - t\right)\]

  10. Applied associate-+l+0.3

    \[\leadsto \mathsf{fma}\left(a - 0.5, \log t, \color{blue}{\left(2 \cdot \log \left(\sqrt[3]{z}\right) + \left(\log \left(x + y\right) + \log \left(\sqrt[3]{z}\right)\right)\right)} - t\right)\]

  11. Simplified0.3

    \[\leadsto \mathsf{fma}\left(a - 0.5, \log t, \left(2 \cdot \log \left(\sqrt[3]{z}\right) + \color{blue}{\left(\log \left(\sqrt[3]{z}\right) + \log \left(x + y\right)\right)}\right) - t\right)\]

  12. Final simplification0.3

    \[\leadsto \mathsf{fma}\left(a - 0.5, \log t, \left(2 \cdot \log \left(\sqrt[3]{z}\right) + \left(\log \left(\sqrt[3]{z}\right) + \log \left(x + y\right)\right)\right) - t\right)\]

Reproduce

herbie shell --seed 2019323 +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))))