Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Average Error: 0.3 → 0.3

Time: 18.7s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a) {
        double r398363 = x;
        double r398364 = y;
        double r398365 = r398363 + r398364;
        double r398366 = log(r398365);
        double r398367 = z;
        double r398368 = log(r398367);
        double r398369 = r398366 + r398368;
        double r398370 = t;
        double r398371 = r398369 - r398370;
        double r398372 = a;
        double r398373 = 0.5;
        double r398374 = r398372 - r398373;
        double r398375 = log(r398370);
        double r398376 = r398374 * r398375;
        double r398377 = r398371 + r398376;
        return r398377;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r398378 = x;
        double r398379 = y;
        double r398380 = r398378 + r398379;
        double r398381 = log(r398380);
        double r398382 = z;
        double r398383 = log(r398382);
        double r398384 = t;
        double r398385 = r398383 - r398384;
        double r398386 = a;
        double r398387 = 0.5;
        double r398388 = r398386 - r398387;
        double r398389 = 2.0;
        double r398390 = cbrt(r398384);
        double r398391 = log(r398390);
        double r398392 = r398389 * r398391;
        double r398393 = r398388 * r398392;
        double r398394 = r398389 * r398388;
        double r398395 = cbrt(r398390);
        double r398396 = log(r398395);
        double r398397 = r398394 * r398396;
        double r398398 = r398396 * r398388;
        double r398399 = r398397 + r398398;
        double r398400 = r398393 + r398399;
        double r398401 = r398385 + r398400;
        double r398402 = r398381 + r398401;
        return r398402;
}

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 associate--l+ 0.3

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

4. Applied associate-+l+ 0.3

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

6. Applied add-cube-cbrt 0.3

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

7. Applied log-prod 0.3

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

8. Applied distribute-lft-in 0.3

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

9. Simplified 0.3

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

11. Applied add-cube-cbrt 0.3

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

12. Applied log-prod 0.3

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

13. Applied distribute-lft-in 0.3

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

14. Simplified 0.3

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

15. Simplified 0.3

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

16. Final simplification 0.3

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

Reproduce

herbie shell --seed 2020047 
(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))))