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

Average Error: 15.8 → 0.3

Time: 17.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -5.435048613210959909118539310961078344525 \cdot 10^{-309}:\\ \;\;\;\;x \cdot \left(\log \left(\frac{-1}{y}\right) - \log \left(\frac{-1}{x}\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) + x \cdot \log \left(\frac{\sqrt[3]{{x}^{\frac{2}{3}}} \cdot \sqrt[3]{\sqrt[3]{x}}}{\sqrt[3]{y}}\right)\right) - z\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r278790 = x;
        double r278791 = y;
        double r278792 = r278790 / r278791;
        double r278793 = log(r278792);
        double r278794 = r278790 * r278793;
        double r278795 = z;
        double r278796 = r278794 - r278795;
        return r278796;
}

↓

double f(double x, double y, double z) {
        double r278797 = y;
        double r278798 = -5.43504861321096e-309;
        bool r278799 = r278797 <= r278798;
        double r278800 = x;
        double r278801 = -1.0;
        double r278802 = r278801 / r278797;
        double r278803 = log(r278802);
        double r278804 = r278801 / r278800;
        double r278805 = log(r278804);
        double r278806 = r278803 - r278805;
        double r278807 = r278800 * r278806;
        double r278808 = z;
        double r278809 = r278807 - r278808;
        double r278810 = 2.0;
        double r278811 = cbrt(r278800);
        double r278812 = cbrt(r278797);
        double r278813 = r278811 / r278812;
        double r278814 = log(r278813);
        double r278815 = r278810 * r278814;
        double r278816 = r278800 * r278815;
        double r278817 = 0.6666666666666666;
        double r278818 = pow(r278800, r278817);
        double r278819 = cbrt(r278818);
        double r278820 = cbrt(r278811);
        double r278821 = r278819 * r278820;
        double r278822 = r278821 / r278812;
        double r278823 = log(r278822);
        double r278824 = r278800 * r278823;
        double r278825 = r278816 + r278824;
        double r278826 = r278825 - r278808;
        double r278827 = r278799 ? r278809 : r278826;
        return r278827;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	15.8
Target	8.2
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. Split input into 2 regimes

2. if y < -5.43504861321096e-309

   1. Initial program 16.3

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

   2. Taylor expanded around -inf 0.3

      \[\leadsto x \cdot \color{blue}{\left(\log \left(\frac{-1}{y}\right) - \log \left(\frac{-1}{x}\right)\right)} - z\]

   if -5.43504861321096e-309 < y

   1. Initial program 15.2

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

   3. Applied add-cube-cbrt 15.2

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

   4. Applied add-cube-cbrt 15.2

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

   5. Applied times-frac 15.2

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

   6. Applied log-prod 3.7

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

   7. Applied distribute-lft-in 3.7

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

   8. Simplified 0.2

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

   10. Applied add-cube-cbrt 0.2

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

   11. Applied cbrt-prod 0.2

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

   12. Simplified 0.2

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -5.435048613210959909118539310961078344525 \cdot 10^{-309}:\\ \;\;\;\;x \cdot \left(\log \left(\frac{-1}{y}\right) - \log \left(\frac{-1}{x}\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) + x \cdot \log \left(\frac{\sqrt[3]{{x}^{\frac{2}{3}}} \cdot \sqrt[3]{\sqrt[3]{x}}}{\sqrt[3]{y}}\right)\right) - z\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +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 7.595077799083773e-308) (- (* x (log (/ x y))) z) (- (* x (- (log x) (log y))) z))

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