Numeric.SpecFunctions:logBeta from math-functions-0.1.5.2, B

Average Error: 0.1 → 0.1

Time: 10.6s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r81925 = x;
        double r81926 = y;
        double r81927 = log(r81926);
        double r81928 = r81925 * r81927;
        double r81929 = z;
        double r81930 = r81928 + r81929;
        double r81931 = t;
        double r81932 = r81930 + r81931;
        double r81933 = a;
        double r81934 = r81932 + r81933;
        double r81935 = b;
        double r81936 = 0.5;
        double r81937 = r81935 - r81936;
        double r81938 = c;
        double r81939 = log(r81938);
        double r81940 = r81937 * r81939;
        double r81941 = r81934 + r81940;
        double r81942 = i;
        double r81943 = r81926 * r81942;
        double r81944 = r81941 + r81943;
        return r81944;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r81945 = 0.6666666666666666;
        double r81946 = y;
        double r81947 = log(r81946);
        double r81948 = x;
        double r81949 = r81947 * r81948;
        double r81950 = r81945 * r81949;
        double r81951 = cbrt(r81946);
        double r81952 = log(r81951);
        double r81953 = r81948 * r81952;
        double r81954 = r81950 + r81953;
        double r81955 = z;
        double r81956 = r81954 + r81955;
        double r81957 = t;
        double r81958 = r81956 + r81957;
        double r81959 = a;
        double r81960 = r81958 + r81959;
        double r81961 = b;
        double r81962 = 0.5;
        double r81963 = r81961 - r81962;
        double r81964 = c;
        double r81965 = log(r81964);
        double r81966 = r81963 * r81965;
        double r81967 = r81960 + r81966;
        double r81968 = i;
        double r81969 = r81946 * r81968;
        double r81970 = r81967 + r81969;
        return r81970;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus i

Derivation

1. Initial program 0.1

   \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\]
2. Using strategy rm

3. Applied add-cube-cbrt 0.1

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

4. Applied log-prod 0.1

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

5. Applied distribute-lft-in 0.1

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

6. Simplified 0.1

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

8. Applied *-un-lft-identity 0.1

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

9. Applied cbrt-prod 0.1

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

10. Applied log-prod 0.1

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

11. Applied distribute-lft-in 0.1

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

12. Applied associate-+r+ 0.1

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

13. Simplified 0.1

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

14. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019356 
(FPCore (x y z t a b c i)
  :name "Numeric.SpecFunctions:logBeta from math-functions-0.1.5.2, B"
  :precision binary64
  (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i)))