Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Average Error: 0.3 → 0.3

Time: 18.3s

Precision: 64

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

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a) {
        double r249079 = x;
        double r249080 = y;
        double r249081 = r249079 + r249080;
        double r249082 = log(r249081);
        double r249083 = z;
        double r249084 = log(r249083);
        double r249085 = r249082 + r249084;
        double r249086 = t;
        double r249087 = r249085 - r249086;
        double r249088 = a;
        double r249089 = 0.5;
        double r249090 = r249088 - r249089;
        double r249091 = log(r249086);
        double r249092 = r249090 * r249091;
        double r249093 = r249087 + r249092;
        return r249093;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r249094 = x;
        double r249095 = y;
        double r249096 = r249094 + r249095;
        double r249097 = log(r249096);
        double r249098 = z;
        double r249099 = sqrt(r249098);
        double r249100 = log(r249099);
        double r249101 = r249097 + r249100;
        double r249102 = a;
        double r249103 = 0.5;
        double r249104 = r249102 - r249103;
        double r249105 = 2.0;
        double r249106 = t;
        double r249107 = 0.6666666666666666;
        double r249108 = pow(r249106, r249107);
        double r249109 = cbrt(r249108);
        double r249110 = cbrt(r249106);
        double r249111 = cbrt(r249110);
        double r249112 = r249109 * r249111;
        double r249113 = log(r249112);
        double r249114 = r249105 * r249113;
        double r249115 = r249104 * r249114;
        double r249116 = r249100 - r249106;
        double r249117 = r249115 + r249116;
        double r249118 = log(r249110);
        double r249119 = r249118 * r249104;
        double r249120 = r249117 + r249119;
        double r249121 = r249101 + r249120;
        return r249121;
}

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 add-sqr-sqrt 0.3

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

4. Applied log-prod 0.3

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

5. Applied associate-+r+ 0.3

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

7. Applied associate--l+ 0.3

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

8. Applied associate-+l+ 0.3

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

10. Applied add-cube-cbrt 0.3

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

11. Applied log-prod 0.3

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

12. Applied distribute-rgt-in 0.3

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

13. Applied associate-+r+ 0.3

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

14. Simplified 0.3

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

16. Applied add-cube-cbrt 0.3

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

17. Applied cbrt-prod 0.3

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

18. Simplified 0.3

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

19. Final simplification 0.3

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

Reproduce

herbie shell --seed 2020046 
(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))))