Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Average Error: 0.3 → 0.3

Time: 32.9s

Precision: 64

Math

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

↓

\[\log \left(x + y\right) - \left(\left(t - \log z\right) - \left(\left(a - 0.5\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{t}\right)\right) + \left(a - 0.5\right) \cdot \left(\frac{1}{3} \cdot \log t\right)\right)\right)\]

C

double f(double x, double y, double z, double t, double a) {
        double r276856 = x;
        double r276857 = y;
        double r276858 = r276856 + r276857;
        double r276859 = log(r276858);
        double r276860 = z;
        double r276861 = log(r276860);
        double r276862 = r276859 + r276861;
        double r276863 = t;
        double r276864 = r276862 - r276863;
        double r276865 = a;
        double r276866 = 0.5;
        double r276867 = r276865 - r276866;
        double r276868 = log(r276863);
        double r276869 = r276867 * r276868;
        double r276870 = r276864 + r276869;
        return r276870;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r276871 = x;
        double r276872 = y;
        double r276873 = r276871 + r276872;
        double r276874 = log(r276873);
        double r276875 = t;
        double r276876 = z;
        double r276877 = log(r276876);
        double r276878 = r276875 - r276877;
        double r276879 = a;
        double r276880 = 0.5;
        double r276881 = r276879 - r276880;
        double r276882 = 2.0;
        double r276883 = cbrt(r276875);
        double r276884 = log(r276883);
        double r276885 = r276882 * r276884;
        double r276886 = r276881 * r276885;
        double r276887 = 0.3333333333333333;
        double r276888 = log(r276875);
        double r276889 = r276887 * r276888;
        double r276890 = r276881 * r276889;
        double r276891 = r276886 + r276890;
        double r276892 = r276878 - r276891;
        double r276893 = r276874 - r276892;
        return r276893;
}

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. Simplified 0.3

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

4. Applied add-cube-cbrt 0.3

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

5. Applied log-prod 0.3

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

6. Applied distribute-lft-in 0.3

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

7. Simplified 0.3

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

8. Taylor expanded around inf 0.3

   \[\leadsto \log \left(y + x\right) - \left(\left(t - \log z\right) - \left(\left(2 \cdot \log \left(\sqrt[3]{t}\right)\right) \cdot \left(a - 0.5\right) + \color{blue}{\left(a \cdot \log \left({\left(\frac{1}{t}\right)}^{\frac{-1}{3}}\right) - 0.5 \cdot \log \left({\left(\frac{1}{t}\right)}^{\frac{-1}{3}}\right)\right)}\right)\right)\]

9. Simplified 0.3

   \[\leadsto \log \left(y + x\right) - \left(\left(t - \log z\right) - \left(\left(2 \cdot \log \left(\sqrt[3]{t}\right)\right) \cdot \left(a - 0.5\right) + \color{blue}{\left(a - 0.5\right) \cdot \left(-\frac{-1}{3} \cdot \log t\right)}\right)\right)\]

10. Final simplification 0.3

    \[\leadsto \log \left(x + y\right) - \left(\left(t - \log z\right) - \left(\left(a - 0.5\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{t}\right)\right) + \left(a - 0.5\right) \cdot \left(\frac{1}{3} \cdot \log t\right)\right)\right)\]

Reproduce

herbie shell --seed 2019194 
(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))))