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

Average Error: 0.1 → 0.1

Time: 11.0s

Precision: 64

Math

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

↓

\[\left(\left(\left(\left(\mathsf{fma}\left(x, 2 \cdot \log \left(\sqrt[3]{y}\right), {\left(x \cdot \log \left({y}^{\frac{1}{3}}\right)\right)}^{1}\right) + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r72731 = x;
        double r72732 = y;
        double r72733 = log(r72732);
        double r72734 = r72731 * r72733;
        double r72735 = z;
        double r72736 = r72734 + r72735;
        double r72737 = t;
        double r72738 = r72736 + r72737;
        double r72739 = a;
        double r72740 = r72738 + r72739;
        double r72741 = b;
        double r72742 = 0.5;
        double r72743 = r72741 - r72742;
        double r72744 = c;
        double r72745 = log(r72744);
        double r72746 = r72743 * r72745;
        double r72747 = r72740 + r72746;
        double r72748 = i;
        double r72749 = r72732 * r72748;
        double r72750 = r72747 + r72749;
        return r72750;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r72751 = x;
        double r72752 = 2.0;
        double r72753 = y;
        double r72754 = cbrt(r72753);
        double r72755 = log(r72754);
        double r72756 = r72752 * r72755;
        double r72757 = 0.3333333333333333;
        double r72758 = pow(r72753, r72757);
        double r72759 = log(r72758);
        double r72760 = r72751 * r72759;
        double r72761 = 1.0;
        double r72762 = pow(r72760, r72761);
        double r72763 = fma(r72751, r72756, r72762);
        double r72764 = z;
        double r72765 = r72763 + r72764;
        double r72766 = t;
        double r72767 = r72765 + r72766;
        double r72768 = a;
        double r72769 = r72767 + r72768;
        double r72770 = b;
        double r72771 = 0.5;
        double r72772 = r72770 - r72771;
        double r72773 = c;
        double r72774 = log(r72773);
        double r72775 = r72772 * r72774;
        double r72776 = r72769 + r72775;
        double r72777 = i;
        double r72778 = r72753 * r72777;
        double r72779 = r72776 + r72778;
        return r72779;
}

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 pow1 0.1

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

9. Applied pow1 0.1

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

10. Applied pow-prod-down 0.1

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

11. Simplified 0.1

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

13. Applied fma-def 0.1

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

14. Final simplification 0.1

    \[\leadsto \left(\left(\left(\left(\mathsf{fma}\left(x, 2 \cdot \log \left(\sqrt[3]{y}\right), {\left(x \cdot \log \left({y}^{\frac{1}{3}}\right)\right)}^{1}\right) + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\]

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(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)))