Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Average Error: 0.3 → 0.3

Time: 47.0s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a) {
        double r2582158 = x;
        double r2582159 = y;
        double r2582160 = r2582158 + r2582159;
        double r2582161 = log(r2582160);
        double r2582162 = z;
        double r2582163 = log(r2582162);
        double r2582164 = r2582161 + r2582163;
        double r2582165 = t;
        double r2582166 = r2582164 - r2582165;
        double r2582167 = a;
        double r2582168 = 0.5;
        double r2582169 = r2582167 - r2582168;
        double r2582170 = log(r2582165);
        double r2582171 = r2582169 * r2582170;
        double r2582172 = r2582166 + r2582171;
        return r2582172;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r2582173 = z;
        double r2582174 = sqrt(r2582173);
        double r2582175 = log(r2582174);
        double r2582176 = 0.5;
        double r2582177 = a;
        double r2582178 = r2582176 - r2582177;
        double r2582179 = t;
        double r2582180 = cbrt(r2582179);
        double r2582181 = log(r2582180);
        double r2582182 = r2582178 * r2582181;
        double r2582183 = r2582182 + r2582182;
        double r2582184 = r2582175 - r2582183;
        double r2582185 = r2582184 - r2582182;
        double r2582186 = r2582185 + r2582175;
        double r2582187 = y;
        double r2582188 = x;
        double r2582189 = r2582187 + r2582188;
        double r2582190 = log(r2582189);
        double r2582191 = r2582179 - r2582190;
        double r2582192 = r2582186 - r2582191;
        return r2582192;
}

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. Simplified 0.3

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

4. Applied add-sqr-sqrt 0.3

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

5. Applied log-prod 0.3

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

6. Applied associate--l+ 0.3

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

7. Applied associate-+l+ 0.3

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

9. Applied fma-udef 0.3

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

10. Applied associate--r+ 0.3

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

11. Applied associate-+l- 0.3

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

12. Applied associate-+r- 0.3

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

14. Applied add-cube-cbrt 0.3

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

15. Applied log-prod 0.3

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

16. Applied distribute-rgt-in 0.3

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

17. Applied associate--r+ 0.3

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

18. Simplified 0.3

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

19. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))