Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Average Error: 0.3 → 0.3

Time: 28.8s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a) {
        double r271023 = x;
        double r271024 = y;
        double r271025 = r271023 + r271024;
        double r271026 = log(r271025);
        double r271027 = z;
        double r271028 = log(r271027);
        double r271029 = r271026 + r271028;
        double r271030 = t;
        double r271031 = r271029 - r271030;
        double r271032 = a;
        double r271033 = 0.5;
        double r271034 = r271032 - r271033;
        double r271035 = log(r271030);
        double r271036 = r271034 * r271035;
        double r271037 = r271031 + r271036;
        return r271037;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r271038 = x;
        double r271039 = y;
        double r271040 = r271038 + r271039;
        double r271041 = log(r271040);
        double r271042 = 2.0;
        double r271043 = z;
        double r271044 = cbrt(r271043);
        double r271045 = log(r271044);
        double r271046 = r271042 * r271045;
        double r271047 = r271041 + r271046;
        double r271048 = t;
        double r271049 = r271045 - r271048;
        double r271050 = r271047 + r271049;
        double r271051 = cbrt(r271048);
        double r271052 = log(r271051);
        double r271053 = r271042 * r271052;
        double r271054 = a;
        double r271055 = 0.5;
        double r271056 = r271054 - r271055;
        double r271057 = r271053 * r271056;
        double r271058 = r271052 * r271056;
        double r271059 = r271057 + r271058;
        double r271060 = r271050 + r271059;
        return r271060;
}

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 associate--l+ 0.3

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

5. Applied add-cube-cbrt 0.3

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

6. Applied log-prod 0.3

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

7. Applied distribute-lft-in 0.3

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

8. Simplified 0.3

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

9. Simplified 0.3

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

11. Applied add-cube-cbrt 0.3

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

12. Applied log-prod 0.3

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

13. Applied associate--l+ 0.3

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

14. Applied associate-+r+ 0.3

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

15. Simplified 0.3

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

16. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019209 
(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))))