Numeric.SpecFunctions:logBeta from math-functions-0.1.5.2, A

Average Error: 0.1 → 0.1

Time: 15.6s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a, double b) {
        double r383373 = x;
        double r383374 = y;
        double r383375 = r383373 + r383374;
        double r383376 = z;
        double r383377 = r383375 + r383376;
        double r383378 = t;
        double r383379 = log(r383378);
        double r383380 = r383376 * r383379;
        double r383381 = r383377 - r383380;
        double r383382 = a;
        double r383383 = 0.5;
        double r383384 = r383382 - r383383;
        double r383385 = b;
        double r383386 = r383384 * r383385;
        double r383387 = r383381 + r383386;
        return r383387;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r383388 = z;
        double r383389 = t;
        double r383390 = sqrt(r383389);
        double r383391 = log(r383390);
        double r383392 = cbrt(r383389);
        double r383393 = r383392 * r383392;
        double r383394 = sqrt(r383393);
        double r383395 = log(r383394);
        double r383396 = r383391 + r383395;
        double r383397 = sqrt(r383392);
        double r383398 = log(r383397);
        double r383399 = r383396 + r383398;
        double r383400 = r383388 * r383399;
        double r383401 = a;
        double r383402 = 0.5;
        double r383403 = r383401 - r383402;
        double r383404 = b;
        double r383405 = r383403 * r383404;
        double r383406 = r383400 - r383405;
        double r383407 = r383388 - r383406;
        double r383408 = x;
        double r383409 = y;
        double r383410 = r383408 + r383409;
        double r383411 = r383407 + r383410;
        return r383411;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Target

Original	0.1
Target	0.4
Herbie	0.1

\[\left(\left(x + y\right) + \frac{\left(1 - {\left(\log t\right)}^{2}\right) \cdot z}{1 + \log t}\right) + \left(a - 0.5\right) \cdot b\]

Derivation

1. Initial program 0.1

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

3. Applied associate--l+ 0.1

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

5. Applied add-sqr-sqrt 0.1

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

6. Applied log-prod 0.1

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

7. Applied distribute-lft-in 0.1

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

8. Applied associate--r+ 0.1

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

9. Simplified 0.1

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

11. Applied add-cube-cbrt 0.1

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

12. Applied sqrt-prod 0.1

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

13. Applied log-prod 0.1

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

14. Applied distribute-rgt-in 0.1

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

15. Applied associate--r+ 0.1

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

16. Simplified 0.1

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

17. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019294 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:logBeta from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (+ (+ (+ x y) (/ (* (- 1 (pow (log t) 2)) z) (+ 1 (log t)))) (* (- a 0.5) b))

  (+ (- (+ (+ x y) z) (* z (log t))) (* (- a 0.5) b)))