Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Average Error: 0.3 → 0.3

Time: 2.9m

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a) {
        double r739445 = x;
        double r739446 = y;
        double r739447 = r739445 + r739446;
        double r739448 = log(r739447);
        double r739449 = z;
        double r739450 = log(r739449);
        double r739451 = r739448 + r739450;
        double r739452 = t;
        double r739453 = r739451 - r739452;
        double r739454 = a;
        double r739455 = 0.5;
        double r739456 = r739454 - r739455;
        double r739457 = log(r739452);
        double r739458 = r739456 * r739457;
        double r739459 = r739453 + r739458;
        return r739459;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r739460 = a;
        double r739461 = 0.5;
        double r739462 = r739460 - r739461;
        double r739463 = t;
        double r739464 = log(r739463);
        double r739465 = x;
        double r739466 = y;
        double r739467 = r739465 + r739466;
        double r739468 = sqrt(r739467);
        double r739469 = cbrt(r739468);
        double r739470 = r739469 * r739469;
        double r739471 = log(r739470);
        double r739472 = log(r739469);
        double r739473 = log(r739468);
        double r739474 = z;
        double r739475 = log(r739474);
        double r739476 = r739473 + r739475;
        double r739477 = r739472 + r739476;
        double r739478 = r739471 + r739477;
        double r739479 = r739478 - r739463;
        double r739480 = fma(r739462, r739464, r739479);
        return r739480;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	0.3
Target	0.3
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 0.3

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

2. Simplified 0.2

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

4. Applied add-sqr-sqrt 0.2

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

5. Applied log-prod 0.2

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

6. Applied associate-+l+ 0.3

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

8. Applied add-cube-cbrt 0.3

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

9. Applied log-prod 0.3

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

10. Applied associate-+l+ 0.3

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

11. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019305 +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))))