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

Average Error: 15.8 → 0.3

Time: 19.5s

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 r331616 = x;
        double r331617 = y;
        double r331618 = r331616 / r331617;
        double r331619 = log(r331618);
        double r331620 = r331616 * r331619;
        double r331621 = z;
        double r331622 = r331620 - r331621;
        return r331622;
}

↓

double f(double x, double y, double z) {
        double r331623 = y;
        double r331624 = -5.43504861321096e-309;
        bool r331625 = r331623 <= r331624;
        double r331626 = x;
        double r331627 = -1.0;
        double r331628 = r331627 / r331623;
        double r331629 = log(r331628);
        double r331630 = r331627 / r331626;
        double r331631 = log(r331630);
        double r331632 = r331629 - r331631;
        double r331633 = r331626 * r331632;
        double r331634 = z;
        double r331635 = r331633 - r331634;
        double r331636 = 2.0;
        double r331637 = cbrt(r331626);
        double r331638 = cbrt(r331623);
        double r331639 = r331637 / r331638;
        double r331640 = log(r331639);
        double r331641 = r331636 * r331640;
        double r331642 = r331626 * r331641;
        double r331643 = 0.6666666666666666;
        double r331644 = pow(r331626, r331643);
        double r331645 = cbrt(r331644);
        double r331646 = cbrt(r331637);
        double r331647 = r331645 * r331646;
        double r331648 = r331647 / r331638;
        double r331649 = log(r331648);
        double r331650 = r331626 * r331649;
        double r331651 = r331642 + r331650;
        double r331652 = r331651 - r331634;
        double r331653 = r331625 ? r331635 : r331652;
        return r331653;
}

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))