Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Average Error: 0.2 → 0.3

Time: 38.0s

Precision: 64

Math

\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]

↓

\[\left(\left(\left(\left(\log \left(x + y\right) + \log \left(\sqrt{z}\right)\right) + \log \left(\sqrt{\sqrt{z}}\right)\right) + \log \left(\sqrt{\sqrt{z}}\right)\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]

C

double f(double x, double y, double z, double t, double a) {
        double r240440 = x;
        double r240441 = y;
        double r240442 = r240440 + r240441;
        double r240443 = log(r240442);
        double r240444 = z;
        double r240445 = log(r240444);
        double r240446 = r240443 + r240445;
        double r240447 = t;
        double r240448 = r240446 - r240447;
        double r240449 = a;
        double r240450 = 0.5;
        double r240451 = r240449 - r240450;
        double r240452 = log(r240447);
        double r240453 = r240451 * r240452;
        double r240454 = r240448 + r240453;
        return r240454;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r240455 = x;
        double r240456 = y;
        double r240457 = r240455 + r240456;
        double r240458 = log(r240457);
        double r240459 = z;
        double r240460 = sqrt(r240459);
        double r240461 = log(r240460);
        double r240462 = r240458 + r240461;
        double r240463 = sqrt(r240460);
        double r240464 = log(r240463);
        double r240465 = r240462 + r240464;
        double r240466 = r240465 + r240464;
        double r240467 = t;
        double r240468 = r240466 - r240467;
        double r240469 = a;
        double r240470 = 0.5;
        double r240471 = r240469 - r240470;
        double r240472 = log(r240467);
        double r240473 = r240471 * r240472;
        double r240474 = r240468 + r240473;
        return r240474;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	0.2
Target	0.2
Herbie	0.3

\[\log \left(x + y\right) + \left(\left(\log z - t\right) + \left(a - 0.5\right) \cdot \log t\right)\]

Derivation

1. Initial program 0.2

   \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
2. Using strategy rm

3. Applied add-sqr-sqrt 0.2

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

4. Applied log-prod 0.2

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

5. Applied associate-+r+ 0.3

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

7. Applied add-sqr-sqrt 0.3

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

8. Applied sqrt-prod 0.3

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

9. Applied log-prod 0.3

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

10. Applied associate-+r+ 0.3

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

11. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019198 +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))))