Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Average Error: 0.3 → 0.3

Time: 40.4s

Precision: 64

Math

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

↓

\[\left(\log \left({t}^{\frac{1}{3}}\right) \cdot \left(a - 0.5\right) + \left(\log \left(\sqrt[3]{t}\right) + \log \left(\sqrt[3]{t}\right)\right) \cdot \left(a - 0.5\right)\right) + \left(\left(\log \left(\sqrt[3]{y + x} \cdot \sqrt[3]{y + x}\right) + \left(\log z + \log \left(\sqrt[3]{y + x}\right)\right)\right) - t\right)\]

C

double f(double x, double y, double z, double t, double a) {
        double r15034159 = x;
        double r15034160 = y;
        double r15034161 = r15034159 + r15034160;
        double r15034162 = log(r15034161);
        double r15034163 = z;
        double r15034164 = log(r15034163);
        double r15034165 = r15034162 + r15034164;
        double r15034166 = t;
        double r15034167 = r15034165 - r15034166;
        double r15034168 = a;
        double r15034169 = 0.5;
        double r15034170 = r15034168 - r15034169;
        double r15034171 = log(r15034166);
        double r15034172 = r15034170 * r15034171;
        double r15034173 = r15034167 + r15034172;
        return r15034173;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r15034174 = t;
        double r15034175 = 0.3333333333333333;
        double r15034176 = pow(r15034174, r15034175);
        double r15034177 = log(r15034176);
        double r15034178 = a;
        double r15034179 = 0.5;
        double r15034180 = r15034178 - r15034179;
        double r15034181 = r15034177 * r15034180;
        double r15034182 = cbrt(r15034174);
        double r15034183 = log(r15034182);
        double r15034184 = r15034183 + r15034183;
        double r15034185 = r15034184 * r15034180;
        double r15034186 = r15034181 + r15034185;
        double r15034187 = y;
        double r15034188 = x;
        double r15034189 = r15034187 + r15034188;
        double r15034190 = cbrt(r15034189);
        double r15034191 = r15034190 * r15034190;
        double r15034192 = log(r15034191);
        double r15034193 = z;
        double r15034194 = log(r15034193);
        double r15034195 = log(r15034190);
        double r15034196 = r15034194 + r15034195;
        double r15034197 = r15034192 + r15034196;
        double r15034198 = r15034197 - r15034174;
        double r15034199 = r15034186 + r15034198;
        return r15034199;
}

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

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

8. Applied pow1/3 0.3

   \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(\left(a - 0.5\right) \cdot \left(\log \left(\sqrt[3]{t}\right) + \log \left(\sqrt[3]{t}\right)\right) + \left(a - 0.5\right) \cdot \log \color{blue}{\left({t}^{\frac{1}{3}}\right)}\right)\]
9. Using strategy rm

10. Applied add-cube-cbrt 0.3

    \[\leadsto \left(\left(\log \color{blue}{\left(\left(\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}\right) \cdot \sqrt[3]{x + y}\right)} + \log z\right) - t\right) + \left(\left(a - 0.5\right) \cdot \left(\log \left(\sqrt[3]{t}\right) + \log \left(\sqrt[3]{t}\right)\right) + \left(a - 0.5\right) \cdot \log \left({t}^{\frac{1}{3}}\right)\right)\]

11. Applied log-prod 0.3

    \[\leadsto \left(\left(\color{blue}{\left(\log \left(\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}\right) + \log \left(\sqrt[3]{x + y}\right)\right)} + \log z\right) - t\right) + \left(\left(a - 0.5\right) \cdot \left(\log \left(\sqrt[3]{t}\right) + \log \left(\sqrt[3]{t}\right)\right) + \left(a - 0.5\right) \cdot \log \left({t}^{\frac{1}{3}}\right)\right)\]

12. Applied associate-+l+ 0.3

    \[\leadsto \left(\color{blue}{\left(\log \left(\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}\right) + \left(\log \left(\sqrt[3]{x + y}\right) + \log z\right)\right)} - t\right) + \left(\left(a - 0.5\right) \cdot \left(\log \left(\sqrt[3]{t}\right) + \log \left(\sqrt[3]{t}\right)\right) + \left(a - 0.5\right) \cdot \log \left({t}^{\frac{1}{3}}\right)\right)\]

13. Final simplification 0.3

    \[\leadsto \left(\log \left({t}^{\frac{1}{3}}\right) \cdot \left(a - 0.5\right) + \left(\log \left(\sqrt[3]{t}\right) + \log \left(\sqrt[3]{t}\right)\right) \cdot \left(a - 0.5\right)\right) + \left(\left(\log \left(\sqrt[3]{y + x} \cdot \sqrt[3]{y + x}\right) + \left(\log z + \log \left(\sqrt[3]{y + x}\right)\right)\right) - t\right)\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"

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

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