Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Average Error: 0.3 → 0.3

Time: 12.6s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a) {
        double r85729 = x;
        double r85730 = y;
        double r85731 = r85729 + r85730;
        double r85732 = log(r85731);
        double r85733 = z;
        double r85734 = log(r85733);
        double r85735 = r85732 + r85734;
        double r85736 = t;
        double r85737 = r85735 - r85736;
        double r85738 = a;
        double r85739 = 0.5;
        double r85740 = r85738 - r85739;
        double r85741 = log(r85736);
        double r85742 = r85740 * r85741;
        double r85743 = r85737 + r85742;
        return r85743;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r85744 = x;
        double r85745 = y;
        double r85746 = r85744 + r85745;
        double r85747 = log(r85746);
        double r85748 = z;
        double r85749 = sqrt(r85748);
        double r85750 = cbrt(r85749);
        double r85751 = r85750 * r85750;
        double r85752 = log(r85751);
        double r85753 = r85747 + r85752;
        double r85754 = log(r85750);
        double r85755 = r85753 + r85754;
        double r85756 = log(r85749);
        double r85757 = r85755 + r85756;
        double r85758 = t;
        double r85759 = r85757 - r85758;
        double r85760 = a;
        double r85761 = 0.5;
        double r85762 = r85760 - r85761;
        double r85763 = sqrt(r85758);
        double r85764 = log(r85763);
        double r85765 = r85762 * r85764;
        double r85766 = r85765 + r85765;
        double r85767 = r85759 + r85766;
        return r85767;
}

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-sqr-sqrt 0.3

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

4. Applied log-prod 0.3

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

5. Applied distribute-lft-in 0.3

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

7. Applied add-sqr-sqrt 0.3

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

8. Applied log-prod 0.3

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

9. Applied associate-+r+ 0.3

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

11. Applied add-cube-cbrt 0.3

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

12. Applied log-prod 0.3

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

13. Applied associate-+r+ 0.3

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

14. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019353 
(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))))