Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Average Error: 0.3 → 0.3

Time: 15.5s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a) {
        double r407371 = x;
        double r407372 = y;
        double r407373 = r407371 + r407372;
        double r407374 = log(r407373);
        double r407375 = z;
        double r407376 = log(r407375);
        double r407377 = r407374 + r407376;
        double r407378 = t;
        double r407379 = r407377 - r407378;
        double r407380 = a;
        double r407381 = 0.5;
        double r407382 = r407380 - r407381;
        double r407383 = log(r407378);
        double r407384 = r407382 * r407383;
        double r407385 = r407379 + r407384;
        return r407385;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r407386 = x;
        double r407387 = y;
        double r407388 = r407386 + r407387;
        double r407389 = cbrt(r407388);
        double r407390 = r407389 * r407389;
        double r407391 = log(r407390);
        double r407392 = log(r407389);
        double r407393 = z;
        double r407394 = log(r407393);
        double r407395 = r407392 + r407394;
        double r407396 = r407391 + r407395;
        double r407397 = t;
        double r407398 = r407396 - r407397;
        double r407399 = a;
        double r407400 = 0.5;
        double r407401 = r407399 - r407400;
        double r407402 = sqrt(r407397);
        double r407403 = log(r407402);
        double r407404 = r407401 * r407403;
        double r407405 = r407404 + r407404;
        double r407406 = r407398 + r407405;
        return r407406;
}

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. Using strategy rm

3. Applied add-sqr-sqrt 0.3

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

4. Applied log-prod 0.3

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

5. Applied distribute-lft-in 0.3

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

7. Applied add-cube-cbrt 0.3

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

8. Applied log-prod 0.3

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

9. Applied associate-+l+ 0.3

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

10. Final simplification 0.3

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

Reproduce

herbie shell --seed 2020035 
(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))))