Average Error: 15.1 → 0.4
Time: 16.8s
Precision: 64
\[x \cdot \log \left(\frac{x}{y}\right) - z\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -5.640858341756123312303713450472891686377 \cdot 10^{-309}:\\ \;\;\;\;x \cdot \log \left(\frac{-1}{y}\right) - \mathsf{fma}\left(\log \left(\frac{-1}{x}\right), x, z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log x \cdot x + x \cdot \log \left(\frac{1}{y}\right)\right) - z\\ \end{array}\]
x \cdot \log \left(\frac{x}{y}\right) - z
↓
\begin{array}{l}
\mathbf{if}\;y \le -5.640858341756123312303713450472891686377 \cdot 10^{-309}:\\
\;\;\;\;x \cdot \log \left(\frac{-1}{y}\right) - \mathsf{fma}\left(\log \left(\frac{-1}{x}\right), x, z\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\log x \cdot x + x \cdot \log \left(\frac{1}{y}\right)\right) - z\\

\end{array}
double f(double x, double y, double z) {
        double r426565 = x;
        double r426566 = y;
        double r426567 = r426565 / r426566;
        double r426568 = log(r426567);
        double r426569 = r426565 * r426568;
        double r426570 = z;
        double r426571 = r426569 - r426570;
        return r426571;
}
↓
double f(double x, double y, double z) {
        double r426572 = y;
        double r426573 = -5.640858341756123e-309;
        bool r426574 = r426572 <= r426573;
        double r426575 = x;
        double r426576 = -1.0;
        double r426577 = r426576 / r426572;
        double r426578 = log(r426577);
        double r426579 = r426575 * r426578;
        double r426580 = r426576 / r426575;
        double r426581 = log(r426580);
        double r426582 = z;
        double r426583 = fma(r426581, r426575, r426582);
        double r426584 = r426579 - r426583;
        double r426585 = log(r426575);
        double r426586 = r426585 * r426575;
        double r426587 = 1.0;
        double r426588 = r426587 / r426572;
        double r426589 = log(r426588);
        double r426590 = r426575 * r426589;
        double r426591 = r426586 + r426590;
        double r426592 = r426591 - r426582;
        double r426593 = r426574 ? r426584 : r426592;
        return r426593;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original15.1
Target7.9
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;y \lt 7.595077799083772773657101400994168792118 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -5.640858341756123e-309

    1. Initial program 15.2

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

    2. Taylor expanded around -inf 0.4

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

    3. Simplified0.4

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

    if -5.640858341756123e-309 < y

    1. Initial program 15.1

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

    3. Applied *-un-lft-identity15.1

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

    4. Applied add-cube-cbrt15.1

      \[\leadsto x \cdot \log \left(\frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{1 \cdot y}\right) - z\]

    5. Applied times-frac15.1

      \[\leadsto x \cdot \log \color{blue}{\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{1} \cdot \frac{\sqrt[3]{x}}{y}\right)} - z\]

    6. Applied log-prod4.6

      \[\leadsto x \cdot \color{blue}{\left(\log \left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{1}\right) + \log \left(\frac{\sqrt[3]{x}}{y}\right)\right)} - z\]

    7. Applied distribute-lft-in4.7

      \[\leadsto \color{blue}{\left(x \cdot \log \left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{1}\right) + x \cdot \log \left(\frac{\sqrt[3]{x}}{y}\right)\right)} - z\]

    8. Simplified4.7

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

    10. Applied div-inv4.7

      \[\leadsto \left(x \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + x \cdot \log \color{blue}{\left(\sqrt[3]{x} \cdot \frac{1}{y}\right)}\right) - z\]

    11. Applied log-prod0.4

      \[\leadsto \left(x \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + x \cdot \color{blue}{\left(\log \left(\sqrt[3]{x}\right) + \log \left(\frac{1}{y}\right)\right)}\right) - z\]

    12. Applied distribute-lft-in0.4

      \[\leadsto \left(x \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \color{blue}{\left(x \cdot \log \left(\sqrt[3]{x}\right) + x \cdot \log \left(\frac{1}{y}\right)\right)}\right) - z\]

    13. Applied associate-+r+0.4

      \[\leadsto \color{blue}{\left(\left(x \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + x \cdot \log \left(\sqrt[3]{x}\right)\right) + x \cdot \log \left(\frac{1}{y}\right)\right)} - z\]

    14. Simplified0.4

      \[\leadsto \left(\color{blue}{\log x \cdot x} + x \cdot \log \left(\frac{1}{y}\right)\right) - z\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -5.640858341756123312303713450472891686377 \cdot 10^{-309}:\\ \;\;\;\;x \cdot \log \left(\frac{-1}{y}\right) - \mathsf{fma}\left(\log \left(\frac{-1}{x}\right), x, z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log x \cdot x + x \cdot \log \left(\frac{1}{y}\right)\right) - z\\ \end{array}\]

Reproduce

herbie shell --seed 2019208 +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))