Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Average Error: 0.3 → 0.3

Time: 43.8s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a) {
        double r3369132 = x;
        double r3369133 = y;
        double r3369134 = r3369132 + r3369133;
        double r3369135 = log(r3369134);
        double r3369136 = z;
        double r3369137 = log(r3369136);
        double r3369138 = r3369135 + r3369137;
        double r3369139 = t;
        double r3369140 = r3369138 - r3369139;
        double r3369141 = a;
        double r3369142 = 0.5;
        double r3369143 = r3369141 - r3369142;
        double r3369144 = log(r3369139);
        double r3369145 = r3369143 * r3369144;
        double r3369146 = r3369140 + r3369145;
        return r3369146;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r3369147 = t;
        double r3369148 = cbrt(r3369147);
        double r3369149 = sqrt(r3369148);
        double r3369150 = log(r3369149);
        double r3369151 = a;
        double r3369152 = 0.5;
        double r3369153 = r3369151 - r3369152;
        double r3369154 = r3369150 * r3369153;
        double r3369155 = x;
        double r3369156 = y;
        double r3369157 = r3369155 + r3369156;
        double r3369158 = log(r3369157);
        double r3369159 = z;
        double r3369160 = log(r3369159);
        double r3369161 = r3369160 - r3369147;
        double r3369162 = r3369158 + r3369161;
        double r3369163 = sqrt(r3369147);
        double r3369164 = log(r3369163);
        double r3369165 = fabs(r3369148);
        double r3369166 = log(r3369165);
        double r3369167 = r3369164 + r3369166;
        double r3369168 = r3369153 * r3369167;
        double r3369169 = r3369162 + r3369168;
        double r3369170 = r3369154 + r3369169;
        return r3369170;
}

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. Applied associate-+r+ 0.3

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

8. Applied add-cube-cbrt 0.3

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

9. Applied sqrt-prod 0.3

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

10. Applied log-prod 0.3

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

11. Applied distribute-rgt-in 0.3

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

12. Applied associate-+r+ 0.3

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

13. Simplified 0.3

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

14. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019169 
(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))))