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

Average Error: 0.1 → 0.1

Time: 6.6s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a, double b) {
        double r371743 = x;
        double r371744 = y;
        double r371745 = r371743 + r371744;
        double r371746 = z;
        double r371747 = r371745 + r371746;
        double r371748 = t;
        double r371749 = log(r371748);
        double r371750 = r371746 * r371749;
        double r371751 = r371747 - r371750;
        double r371752 = a;
        double r371753 = 0.5;
        double r371754 = r371752 - r371753;
        double r371755 = b;
        double r371756 = r371754 * r371755;
        double r371757 = r371751 + r371756;
        return r371757;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r371758 = x;
        double r371759 = y;
        double r371760 = r371758 + r371759;
        double r371761 = z;
        double r371762 = r371760 + r371761;
        double r371763 = 2.0;
        double r371764 = t;
        double r371765 = cbrt(r371764);
        double r371766 = log(r371765);
        double r371767 = r371763 * r371766;
        double r371768 = r371761 * r371767;
        double r371769 = cbrt(r371765);
        double r371770 = log(r371769);
        double r371771 = r371770 * r371763;
        double r371772 = r371761 * r371771;
        double r371773 = r371761 * r371770;
        double r371774 = r371772 + r371773;
        double r371775 = r371768 + r371774;
        double r371776 = r371762 - r371775;
        double r371777 = a;
        double r371778 = 0.5;
        double r371779 = r371777 - r371778;
        double r371780 = b;
        double r371781 = r371779 * r371780;
        double r371782 = r371776 + r371781;
        return r371782;
}

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

Target

Original	0.1
Target	0.3
Herbie	0.1

\[\left(\left(x + y\right) + \frac{\left(1 - {\left(\log t\right)}^{2}\right) \cdot z}{1 + \log t}\right) + \left(a - 0.5\right) \cdot b\]

Derivation

1. Initial program 0.1

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

3. Applied add-cube-cbrt 0.1

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

4. Applied log-prod 0.1

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

5. Applied distribute-lft-in 0.1

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

6. Simplified 0.1

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

8. Applied add-cube-cbrt 0.1

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

9. Applied log-prod 0.1

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

10. Applied distribute-lft-in 0.1

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

11. Simplified 0.1

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

12. Final simplification 0.1

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

Reproduce

herbie shell --seed 2020062 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:logBeta from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (+ (+ (+ x y) (/ (* (- 1 (pow (log t) 2)) z) (+ 1 (log t)))) (* (- a 0.5) b))

  (+ (- (+ (+ x y) z) (* z (log t))) (* (- a 0.5) b)))