Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Average Error: 0.3 → 0.3

Time: 12.3s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a) {
        double r386919 = x;
        double r386920 = y;
        double r386921 = r386919 + r386920;
        double r386922 = log(r386921);
        double r386923 = z;
        double r386924 = log(r386923);
        double r386925 = r386922 + r386924;
        double r386926 = t;
        double r386927 = r386925 - r386926;
        double r386928 = a;
        double r386929 = 0.5;
        double r386930 = r386928 - r386929;
        double r386931 = log(r386926);
        double r386932 = r386930 * r386931;
        double r386933 = r386927 + r386932;
        return r386933;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r386934 = x;
        double r386935 = y;
        double r386936 = r386934 + r386935;
        double r386937 = cbrt(r386936);
        double r386938 = r386937 * r386937;
        double r386939 = log(r386938);
        double r386940 = log(r386937);
        double r386941 = 2.0;
        double r386942 = z;
        double r386943 = sqrt(r386942);
        double r386944 = log(r386943);
        double r386945 = r386941 * r386944;
        double r386946 = r386940 + r386945;
        double r386947 = r386939 + r386946;
        double r386948 = t;
        double r386949 = sqrt(r386948);
        double r386950 = log(r386949);
        double r386951 = a;
        double r386952 = 0.5;
        double r386953 = r386951 - r386952;
        double r386954 = r386950 * r386953;
        double r386955 = r386948 - r386954;
        double r386956 = r386947 - r386955;
        double r386957 = r386956 + r386954;
        return r386957;
}

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-sqr-sqrt 0.3

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

4. Applied log-prod 0.3

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

5. Applied associate-+r+ 0.3

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

7. Applied add-sqr-sqrt 0.3

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

8. Applied log-prod 0.3

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

9. Applied distribute-rgt-in 0.3

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

10. Applied associate-+r+ 0.3

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

11. Simplified 0.3

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

13. Applied add-cube-cbrt 0.3

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

14. Applied log-prod 0.3

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

15. Applied associate-+l+ 0.3

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

16. Final simplification 0.3

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

Reproduce

herbie shell --seed 2020046 
(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))))