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

Average Error: 0.1 → 0.1

Time: 29.9s

Precision: 64

Math

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

↓

\[\left(\left(x + \left(\left(z + y\right) - \left(2 \cdot \log \left({\left({\left(\sqrt[3]{t}\right)}^{\left(\sqrt{\frac{2}{3}}\right)}\right)}^{\left(\sqrt{\frac{2}{3}}\right)} \cdot \sqrt[3]{\sqrt[3]{t}}\right)\right) \cdot z\right)\right) - z \cdot \log \left({t}^{\frac{1}{3}}\right)\right) + \left(a - 0.5\right) \cdot b\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r199281 = x;
        double r199282 = y;
        double r199283 = r199281 + r199282;
        double r199284 = z;
        double r199285 = r199283 + r199284;
        double r199286 = t;
        double r199287 = log(r199286);
        double r199288 = r199284 * r199287;
        double r199289 = r199285 - r199288;
        double r199290 = a;
        double r199291 = 0.5;
        double r199292 = r199290 - r199291;
        double r199293 = b;
        double r199294 = r199292 * r199293;
        double r199295 = r199289 + r199294;
        return r199295;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r199296 = x;
        double r199297 = z;
        double r199298 = y;
        double r199299 = r199297 + r199298;
        double r199300 = 2.0;
        double r199301 = t;
        double r199302 = cbrt(r199301);
        double r199303 = 0.6666666666666666;
        double r199304 = sqrt(r199303);
        double r199305 = pow(r199302, r199304);
        double r199306 = pow(r199305, r199304);
        double r199307 = cbrt(r199302);
        double r199308 = r199306 * r199307;
        double r199309 = log(r199308);
        double r199310 = r199300 * r199309;
        double r199311 = r199310 * r199297;
        double r199312 = r199299 - r199311;
        double r199313 = r199296 + r199312;
        double r199314 = 0.3333333333333333;
        double r199315 = pow(r199301, r199314);
        double r199316 = log(r199315);
        double r199317 = r199297 * r199316;
        double r199318 = r199313 - r199317;
        double r199319 = a;
        double r199320 = 0.5;
        double r199321 = r199319 - r199320;
        double r199322 = b;
        double r199323 = r199321 * r199322;
        double r199324 = r199318 + r199323;
        return r199324;
}

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. 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

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

9. Applied pow1/3 0.1

   \[\leadsto \left(\left(x + \left(\left(z + y\right) - \left(2 \cdot \log \left(\sqrt[3]{t}\right)\right) \cdot z\right)\right) - z \cdot \log \color{blue}{\left({t}^{\frac{1}{3}}\right)}\right) + \left(a - 0.5\right) \cdot b\]
10. Using strategy rm

11. Applied add-cube-cbrt 0.1

    \[\leadsto \left(\left(x + \left(\left(z + y\right) - \left(2 \cdot \log \color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{t}} \cdot \sqrt[3]{\sqrt[3]{t}}\right) \cdot \sqrt[3]{\sqrt[3]{t}}\right)}\right) \cdot z\right)\right) - z \cdot \log \left({t}^{\frac{1}{3}}\right)\right) + \left(a - 0.5\right) \cdot b\]

12. Simplified 0.1

    \[\leadsto \left(\left(x + \left(\left(z + y\right) - \left(2 \cdot \log \left(\color{blue}{{\left(\sqrt[3]{t}\right)}^{\frac{2}{3}}} \cdot \sqrt[3]{\sqrt[3]{t}}\right)\right) \cdot z\right)\right) - z \cdot \log \left({t}^{\frac{1}{3}}\right)\right) + \left(a - 0.5\right) \cdot b\]
13. Using strategy rm

14. Applied add-sqr-sqrt 0.1

    \[\leadsto \left(\left(x + \left(\left(z + y\right) - \left(2 \cdot \log \left({\left(\sqrt[3]{t}\right)}^{\color{blue}{\left(\sqrt{\frac{2}{3}} \cdot \sqrt{\frac{2}{3}}\right)}} \cdot \sqrt[3]{\sqrt[3]{t}}\right)\right) \cdot z\right)\right) - z \cdot \log \left({t}^{\frac{1}{3}}\right)\right) + \left(a - 0.5\right) \cdot b\]

15. Applied pow-unpow 0.1

    \[\leadsto \left(\left(x + \left(\left(z + y\right) - \left(2 \cdot \log \left(\color{blue}{{\left({\left(\sqrt[3]{t}\right)}^{\left(\sqrt{\frac{2}{3}}\right)}\right)}^{\left(\sqrt{\frac{2}{3}}\right)}} \cdot \sqrt[3]{\sqrt[3]{t}}\right)\right) \cdot z\right)\right) - z \cdot \log \left({t}^{\frac{1}{3}}\right)\right) + \left(a - 0.5\right) \cdot b\]

16. Final simplification 0.1

    \[\leadsto \left(\left(x + \left(\left(z + y\right) - \left(2 \cdot \log \left({\left({\left(\sqrt[3]{t}\right)}^{\left(\sqrt{\frac{2}{3}}\right)}\right)}^{\left(\sqrt{\frac{2}{3}}\right)} \cdot \sqrt[3]{\sqrt[3]{t}}\right)\right) \cdot z\right)\right) - z \cdot \log \left({t}^{\frac{1}{3}}\right)\right) + \left(a - 0.5\right) \cdot b\]

Reproduce

herbie shell --seed 2019326 
(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)))