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

Average Error: 0.1 → 0.1

Time: 6.1s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a, double b) {
        double r315304 = x;
        double r315305 = y;
        double r315306 = r315304 + r315305;
        double r315307 = z;
        double r315308 = r315306 + r315307;
        double r315309 = t;
        double r315310 = log(r315309);
        double r315311 = r315307 * r315310;
        double r315312 = r315308 - r315311;
        double r315313 = a;
        double r315314 = 0.5;
        double r315315 = r315313 - r315314;
        double r315316 = b;
        double r315317 = r315315 * r315316;
        double r315318 = r315312 + r315317;
        return r315318;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r315319 = x;
        double r315320 = y;
        double r315321 = r315319 + r315320;
        double r315322 = z;
        double r315323 = r315321 + r315322;
        double r315324 = 2.0;
        double r315325 = t;
        double r315326 = cbrt(r315325);
        double r315327 = log(r315326);
        double r315328 = r315324 * r315327;
        double r315329 = r315322 * r315328;
        double r315330 = cbrt(r315326);
        double r315331 = log(r315330);
        double r315332 = r315331 * r315324;
        double r315333 = r315322 * r315332;
        double r315334 = r315322 * r315331;
        double r315335 = r315333 + r315334;
        double r315336 = r315329 + r315335;
        double r315337 = r315323 - r315336;
        double r315338 = a;
        double r315339 = 0.5;
        double r315340 = r315338 - r315339;
        double r315341 = b;
        double r315342 = r315340 * r315341;
        double r315343 = r315337 + r315342;
        return r315343;
}

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	0.3
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 add-cube-cbrt 0.1

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

4. Applied log-prod 0.1

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

5. Applied distribute-lft-in 0.1

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

6. Simplified 0.1

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

8. Applied add-cube-cbrt 0.1

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

9. Applied log-prod 0.1

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

10. Applied distribute-lft-in 0.1

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

11. Simplified 0.1

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

12. Final simplification 0.1

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

Reproduce

herbie shell --seed 2020062 
(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)))