Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Average Error: 0.3 → 0.3

Time: 30.6s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a) {
        double r48569 = x;
        double r48570 = y;
        double r48571 = r48569 + r48570;
        double r48572 = log(r48571);
        double r48573 = z;
        double r48574 = log(r48573);
        double r48575 = r48572 + r48574;
        double r48576 = t;
        double r48577 = r48575 - r48576;
        double r48578 = a;
        double r48579 = 0.5;
        double r48580 = r48578 - r48579;
        double r48581 = log(r48576);
        double r48582 = r48580 * r48581;
        double r48583 = r48577 + r48582;
        return r48583;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r48584 = x;
        double r48585 = y;
        double r48586 = r48584 + r48585;
        double r48587 = log(r48586);
        double r48588 = z;
        double r48589 = log(r48588);
        double r48590 = r48587 + r48589;
        double r48591 = t;
        double r48592 = a;
        double r48593 = 0.5;
        double r48594 = r48592 - r48593;
        double r48595 = 2.0;
        double r48596 = cbrt(r48591);
        double r48597 = log(r48596);
        double r48598 = r48595 * r48597;
        double r48599 = r48596 * r48596;
        double r48600 = cbrt(r48599);
        double r48601 = log(r48600);
        double r48602 = r48598 + r48601;
        double r48603 = cbrt(r48596);
        double r48604 = log(r48603);
        double r48605 = r48602 + r48604;
        double r48606 = r48594 * r48605;
        double r48607 = r48591 - r48606;
        double r48608 = r48590 - r48607;
        return r48608;
}

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 add-cube-cbrt 0.3

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

4. Applied log-prod 0.3

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

5. Applied distribute-lft-in 0.3

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

6. Simplified 0.3

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

8. Applied add-cube-cbrt 0.3

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

9. Applied cbrt-prod 0.3

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

10. Applied log-prod 0.3

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

11. Applied distribute-lft-in 0.3

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

12. Applied associate-+r+ 0.3

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

13. Simplified 0.3

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

14. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019294 
(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))))