Numeric.SpecFunctions.Extra:bd0 from math-functions-0.1.5.2

Average Error: 15.7 → 0.3

Time: 14.6s

Precision: 64

Math

\[x \cdot \log \left(\frac{x}{y}\right) - z\]

↓

\[x \cdot \left(\sqrt[3]{2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)} \cdot \sqrt[3]{{\left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right)}^{2}}\right) + \left(x \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) - z\right)\]

C

double f(double x, double y, double z) {
        double r291991 = x;
        double r291992 = y;
        double r291993 = r291991 / r291992;
        double r291994 = log(r291993);
        double r291995 = r291991 * r291994;
        double r291996 = z;
        double r291997 = r291995 - r291996;
        return r291997;
}

↓

double f(double x, double y, double z) {
        double r291998 = x;
        double r291999 = 2.0;
        double r292000 = cbrt(r291998);
        double r292001 = y;
        double r292002 = cbrt(r292001);
        double r292003 = r292000 / r292002;
        double r292004 = log(r292003);
        double r292005 = r291999 * r292004;
        double r292006 = cbrt(r292005);
        double r292007 = pow(r292005, r291999);
        double r292008 = cbrt(r292007);
        double r292009 = r292006 * r292008;
        double r292010 = r291998 * r292009;
        double r292011 = r291998 * r292004;
        double r292012 = z;
        double r292013 = r292011 - r292012;
        double r292014 = r292010 + r292013;
        return r292014;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	15.7
Target	8.1
Herbie	0.3

\[\begin{array}{l} \mathbf{if}\;y \lt 7.595077799083772773657101400994168792118 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array}\]

Derivation

1. Initial program 15.7

   \[x \cdot \log \left(\frac{x}{y}\right) - z\]
2. Using strategy rm

3. Applied add-cube-cbrt 15.7

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

4. Applied add-cube-cbrt 15.7

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

5. Applied times-frac 15.7

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

6. Applied log-prod 3.5

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

7. Applied distribute-lft-in 3.5

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

8. Applied associate--l+ 3.5

   \[\leadsto \color{blue}{x \cdot \log \left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) + \left(x \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) - z\right)}\]
9. Using strategy rm

10. Applied add-cbrt-cube 3.5

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

11. Simplified 0.3

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

13. Applied cube-mult 0.3

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

14. Applied cbrt-prod 0.3

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

15. Simplified 0.3

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

16. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019305 +o rules:numerics
(FPCore (x y z)
  :name "Numeric.SpecFunctions.Extra:bd0 from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< y 7.59507779908377277e-308) (- (* x (log (/ x y))) z) (- (* x (- (log x) (log y))) z))

  (- (* x (log (/ x y))) z))