Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Average Error: 0.2 → 0.3

Time: 14.4s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a) {
        double r352016 = x;
        double r352017 = y;
        double r352018 = r352016 + r352017;
        double r352019 = log(r352018);
        double r352020 = z;
        double r352021 = log(r352020);
        double r352022 = r352019 + r352021;
        double r352023 = t;
        double r352024 = r352022 - r352023;
        double r352025 = a;
        double r352026 = 0.5;
        double r352027 = r352025 - r352026;
        double r352028 = log(r352023);
        double r352029 = r352027 * r352028;
        double r352030 = r352024 + r352029;
        return r352030;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r352031 = x;
        double r352032 = y;
        double r352033 = r352031 + r352032;
        double r352034 = log(r352033);
        double r352035 = z;
        double r352036 = log(r352035);
        double r352037 = r352034 + r352036;
        double r352038 = t;
        double r352039 = r352037 - r352038;
        double r352040 = cbrt(r352038);
        double r352041 = r352040 * r352040;
        double r352042 = log(r352041);
        double r352043 = a;
        double r352044 = 0.5;
        double r352045 = r352043 - r352044;
        double r352046 = r352042 * r352045;
        double r352047 = r352039 + r352046;
        double r352048 = 1.0;
        double r352049 = r352048 / r352038;
        double r352050 = -0.3333333333333333;
        double r352051 = pow(r352049, r352050);
        double r352052 = log(r352051);
        double r352053 = r352045 * r352052;
        double r352054 = r352047 + r352053;
        return r352054;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	0.2
Target	0.2
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 0.2

   \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
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. Applied associate-+r+ 0.3

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

7. Simplified 0.3

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

8. Taylor expanded around inf 0.3

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

9. Final simplification 0.3

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

Reproduce

herbie shell --seed 2020083 
(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))))