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

Average Error: 0.1 → 0.1

Time: 7.9s

Precision: 64

Math

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

↓

\[\left(\left(\left(\left(x + y\right) + z\right) - \log \left(\sqrt[3]{\sqrt{t}} \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{\sqrt{t}}\right)\right) \cdot z\right) - z \cdot \log \left({t}^{\frac{1}{3}}\right)\right) + \left(a - 0.5\right) \cdot b\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r249269 = x;
        double r249270 = y;
        double r249271 = r249269 + r249270;
        double r249272 = z;
        double r249273 = r249271 + r249272;
        double r249274 = t;
        double r249275 = log(r249274);
        double r249276 = r249272 * r249275;
        double r249277 = r249273 - r249276;
        double r249278 = a;
        double r249279 = 0.5;
        double r249280 = r249278 - r249279;
        double r249281 = b;
        double r249282 = r249280 * r249281;
        double r249283 = r249277 + r249282;
        return r249283;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r249284 = x;
        double r249285 = y;
        double r249286 = r249284 + r249285;
        double r249287 = z;
        double r249288 = r249286 + r249287;
        double r249289 = t;
        double r249290 = sqrt(r249289);
        double r249291 = cbrt(r249290);
        double r249292 = cbrt(r249289);
        double r249293 = r249292 * r249291;
        double r249294 = r249291 * r249293;
        double r249295 = log(r249294);
        double r249296 = r249295 * r249287;
        double r249297 = r249288 - r249296;
        double r249298 = 0.3333333333333333;
        double r249299 = pow(r249289, r249298);
        double r249300 = log(r249299);
        double r249301 = r249287 * r249300;
        double r249302 = r249297 - r249301;
        double r249303 = a;
        double r249304 = 0.5;
        double r249305 = r249303 - r249304;
        double r249306 = b;
        double r249307 = r249305 * r249306;
        double r249308 = r249302 + r249307;
        return r249308;
}

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 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. Applied associate--r+ 0.1

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

7. Simplified 0.1

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

9. Applied pow1/3 0.1

   \[\leadsto \left(\left(\left(\left(x + y\right) + z\right) - \log \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot z\right) - z \cdot \log \color{blue}{\left({t}^{\frac{1}{3}}\right)}\right) + \left(a - 0.5\right) \cdot b\]
10. Using strategy rm

11. Applied add-sqr-sqrt 0.1

    \[\leadsto \left(\left(\left(\left(x + y\right) + z\right) - \log \left(\sqrt[3]{\color{blue}{\sqrt{t} \cdot \sqrt{t}}} \cdot \sqrt[3]{t}\right) \cdot z\right) - z \cdot \log \left({t}^{\frac{1}{3}}\right)\right) + \left(a - 0.5\right) \cdot b\]

12. Applied cbrt-prod 0.1

    \[\leadsto \left(\left(\left(\left(x + y\right) + z\right) - \log \left(\color{blue}{\left(\sqrt[3]{\sqrt{t}} \cdot \sqrt[3]{\sqrt{t}}\right)} \cdot \sqrt[3]{t}\right) \cdot z\right) - z \cdot \log \left({t}^{\frac{1}{3}}\right)\right) + \left(a - 0.5\right) \cdot b\]

13. Applied associate-*l* 0.1

    \[\leadsto \left(\left(\left(\left(x + y\right) + z\right) - \log \color{blue}{\left(\sqrt[3]{\sqrt{t}} \cdot \left(\sqrt[3]{\sqrt{t}} \cdot \sqrt[3]{t}\right)\right)} \cdot z\right) - z \cdot \log \left({t}^{\frac{1}{3}}\right)\right) + \left(a - 0.5\right) \cdot b\]

14. Simplified 0.1

    \[\leadsto \left(\left(\left(\left(x + y\right) + z\right) - \log \left(\sqrt[3]{\sqrt{t}} \cdot \color{blue}{\left(\sqrt[3]{t} \cdot \sqrt[3]{\sqrt{t}}\right)}\right) \cdot z\right) - z \cdot \log \left({t}^{\frac{1}{3}}\right)\right) + \left(a - 0.5\right) \cdot b\]

15. Final simplification 0.1

    \[\leadsto \left(\left(\left(\left(x + y\right) + z\right) - \log \left(\sqrt[3]{\sqrt{t}} \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{\sqrt{t}}\right)\right) \cdot z\right) - z \cdot \log \left({t}^{\frac{1}{3}}\right)\right) + \left(a - 0.5\right) \cdot b\]

Reproduce

herbie shell --seed 2020003 
(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)))