Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Average Error: 0.3 → 0.3

Time: 30.3s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a) {
        double r66210 = x;
        double r66211 = y;
        double r66212 = r66210 + r66211;
        double r66213 = log(r66212);
        double r66214 = z;
        double r66215 = log(r66214);
        double r66216 = r66213 + r66215;
        double r66217 = t;
        double r66218 = r66216 - r66217;
        double r66219 = a;
        double r66220 = 0.5;
        double r66221 = r66219 - r66220;
        double r66222 = log(r66217);
        double r66223 = r66221 * r66222;
        double r66224 = r66218 + r66223;
        return r66224;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r66225 = x;
        double r66226 = y;
        double r66227 = r66225 + r66226;
        double r66228 = log(r66227);
        double r66229 = 2.0;
        double r66230 = z;
        double r66231 = cbrt(r66230);
        double r66232 = log(r66231);
        double r66233 = r66229 * r66232;
        double r66234 = r66228 + r66233;
        double r66235 = t;
        double r66236 = r66232 - r66235;
        double r66237 = r66234 + r66236;
        double r66238 = cbrt(r66235);
        double r66239 = log(r66238);
        double r66240 = r66229 * r66239;
        double r66241 = a;
        double r66242 = 0.5;
        double r66243 = r66241 - r66242;
        double r66244 = r66240 * r66243;
        double r66245 = r66239 * r66243;
        double r66246 = r66244 + r66245;
        double r66247 = r66237 + r66246;
        return r66247;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

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 associate--l+ 0.3

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

5. Applied add-cube-cbrt 0.3

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

6. Applied log-prod 0.3

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

7. Applied distribute-lft-in 0.3

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

8. Simplified 0.3

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

9. Simplified 0.3

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

11. Applied add-cube-cbrt 0.3

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

12. Applied log-prod 0.3

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

13. Applied associate--l+ 0.3

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

14. Applied associate-+r+ 0.3

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

15. Simplified 0.3

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

16. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019209 
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  :precision binary64
  (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))