Average Error: 0.3 → 0.3
Time: 13.4s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\mathsf{fma}\left(\log t, a - 0.5, \left(\log \left(\sqrt{\sqrt{x + y}}\right) + \left(\log \left(\sqrt{\sqrt{x + y}}\right) + \left(\log \left(\sqrt{x + y}\right) + \log z\right)\right)\right) - t\right)\]
\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t
↓
\mathsf{fma}\left(\log t, a - 0.5, \left(\log \left(\sqrt{\sqrt{x + y}}\right) + \left(\log \left(\sqrt{\sqrt{x + y}}\right) + \left(\log \left(\sqrt{x + y}\right) + \log z\right)\right)\right) - t\right)
double f(double x, double y, double z, double t, double a) {
        double r370445 = x;
        double r370446 = y;
        double r370447 = r370445 + r370446;
        double r370448 = log(r370447);
        double r370449 = z;
        double r370450 = log(r370449);
        double r370451 = r370448 + r370450;
        double r370452 = t;
        double r370453 = r370451 - r370452;
        double r370454 = a;
        double r370455 = 0.5;
        double r370456 = r370454 - r370455;
        double r370457 = log(r370452);
        double r370458 = r370456 * r370457;
        double r370459 = r370453 + r370458;
        return r370459;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r370460 = t;
        double r370461 = log(r370460);
        double r370462 = a;
        double r370463 = 0.5;
        double r370464 = r370462 - r370463;
        double r370465 = x;
        double r370466 = y;
        double r370467 = r370465 + r370466;
        double r370468 = sqrt(r370467);
        double r370469 = sqrt(r370468);
        double r370470 = log(r370469);
        double r370471 = log(r370468);
        double r370472 = z;
        double r370473 = log(r370472);
        double r370474 = r370471 + r370473;
        double r370475 = r370470 + r370474;
        double r370476 = r370470 + r370475;
        double r370477 = r370476 - r370460;
        double r370478 = fma(r370461, r370464, r370477);
        return r370478;
}

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

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

  4. Applied add-sqr-sqrt0.3

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

  5. Applied log-prod0.3

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

  6. Applied associate-+l+0.3

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

  8. Applied add-sqr-sqrt0.3

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

  9. Applied sqrt-prod0.3

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

  10. Applied log-prod0.3

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

  11. Applied associate-+l+0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2020081 +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))))