Average Error: 15.1 → 0.2
Time: 28.4s
Precision: 64
\[x \cdot \log \left(\frac{x}{y}\right) - z\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -5.330202515239102447628611255442331031352 \cdot 10^{-309}:\\ \;\;\;\;\mathsf{fma}\left(x, \log \left(\frac{-1}{y}\right) - \log \left(\frac{-1}{x}\right), -z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \log \left(\frac{\sqrt{x}}{\sqrt{y}}\right) + \log \left(\frac{\sqrt{x}}{\sqrt{y}}\right), -z\right)\\ \end{array}\]
x \cdot \log \left(\frac{x}{y}\right) - z
↓
\begin{array}{l}
\mathbf{if}\;y \le -5.330202515239102447628611255442331031352 \cdot 10^{-309}:\\
\;\;\;\;\mathsf{fma}\left(x, \log \left(\frac{-1}{y}\right) - \log \left(\frac{-1}{x}\right), -z\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(x, \log \left(\frac{\sqrt{x}}{\sqrt{y}}\right) + \log \left(\frac{\sqrt{x}}{\sqrt{y}}\right), -z\right)\\

\end{array}
double f(double x, double y, double z) {
        double r57605475 = x;
        double r57605476 = y;
        double r57605477 = r57605475 / r57605476;
        double r57605478 = log(r57605477);
        double r57605479 = r57605475 * r57605478;
        double r57605480 = z;
        double r57605481 = r57605479 - r57605480;
        return r57605481;
}
↓
double f(double x, double y, double z) {
        double r57605482 = y;
        double r57605483 = -5.3302025152391e-309;
        bool r57605484 = r57605482 <= r57605483;
        double r57605485 = x;
        double r57605486 = -1.0;
        double r57605487 = r57605486 / r57605482;
        double r57605488 = log(r57605487);
        double r57605489 = r57605486 / r57605485;
        double r57605490 = log(r57605489);
        double r57605491 = r57605488 - r57605490;
        double r57605492 = z;
        double r57605493 = -r57605492;
        double r57605494 = fma(r57605485, r57605491, r57605493);
        double r57605495 = sqrt(r57605485);
        double r57605496 = sqrt(r57605482);
        double r57605497 = r57605495 / r57605496;
        double r57605498 = log(r57605497);
        double r57605499 = r57605498 + r57605498;
        double r57605500 = fma(r57605485, r57605499, r57605493);
        double r57605501 = r57605484 ? r57605494 : r57605500;
        return r57605501;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original15.1
Target7.6
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;y \lt 7.595077799083772773657101400994168792118 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -5.3302025152391e-309

    1. Initial program 14.8

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

    3. Applied fma-neg14.8

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

    4. Taylor expanded around -inf 0.3

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

    if -5.3302025152391e-309 < y

    1. Initial program 15.4

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

    3. Applied fma-neg15.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log \left(\frac{x}{y}\right), -z\right)}\]
    4. Using strategy rm

    5. Applied add-sqr-sqrt15.4

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

    6. Applied add-sqr-sqrt15.4

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

    7. Applied times-frac15.4

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

    8. Applied log-prod0.1

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -5.330202515239102447628611255442331031352 \cdot 10^{-309}:\\ \;\;\;\;\mathsf{fma}\left(x, \log \left(\frac{-1}{y}\right) - \log \left(\frac{-1}{x}\right), -z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \log \left(\frac{\sqrt{x}}{\sqrt{y}}\right) + \log \left(\frac{\sqrt{x}}{\sqrt{y}}\right), -z\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019173 +o rules:numerics
(FPCore (x y z)
  :name "Numeric.SpecFunctions.Extra:bd0 from math-functions-0.1.5.2"

  :herbie-target
  (if (< y 7.595077799083773e-308) (- (* x (log (/ x y))) z) (- (* x (- (log x) (log y))) z))

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