Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Average Error: 0.3 → 0.3

Time: 15.4s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a) {
        double r308218 = x;
        double r308219 = y;
        double r308220 = r308218 + r308219;
        double r308221 = log(r308220);
        double r308222 = z;
        double r308223 = log(r308222);
        double r308224 = r308221 + r308223;
        double r308225 = t;
        double r308226 = r308224 - r308225;
        double r308227 = a;
        double r308228 = 0.5;
        double r308229 = r308227 - r308228;
        double r308230 = log(r308225);
        double r308231 = r308229 * r308230;
        double r308232 = r308226 + r308231;
        return r308232;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r308233 = x;
        double r308234 = y;
        double r308235 = r308233 + r308234;
        double r308236 = log(r308235);
        double r308237 = z;
        double r308238 = log(r308237);
        double r308239 = r308236 + r308238;
        double r308240 = t;
        double r308241 = r308239 - r308240;
        double r308242 = a;
        double r308243 = 0.5;
        double r308244 = r308242 - r308243;
        double r308245 = 2.0;
        double r308246 = cbrt(r308240);
        double r308247 = log(r308246);
        double r308248 = r308245 * r308247;
        double r308249 = cbrt(r308246);
        double r308250 = r308249 * r308249;
        double r308251 = log(r308250);
        double r308252 = r308248 + r308251;
        double r308253 = r308244 * r308252;
        double r308254 = log(r308249);
        double r308255 = r308254 * r308244;
        double r308256 = r308253 + r308255;
        double r308257 = r308241 + r308256;
        return r308257;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	0.3
Target	0.3
Herbie	0.3

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

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

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

9. Applied log-prod 0.3

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

10. Applied distribute-rgt-in 0.3

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

11. Applied associate-+r+ 0.3

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

12. Simplified 0.3

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

13. Final simplification 0.3

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

Reproduce

herbie shell --seed 2020036 
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (+ (log (+ x y)) (+ (- (log z) t) (* (- a 0.5) (log t))))

  (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))