Average Error: 15.1 → 0.2
Time: 29.9s
Precision: 64
\[x \cdot \log \left(\frac{x}{y}\right) - z\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -5.330202515239102447628611255442331031352 \cdot 10^{-309}:\\ \;\;\;\;x \cdot \left(\log \left(\frac{-1}{y}\right) - \log \left(\frac{-1}{x}\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \log \left(\frac{\sqrt{x}}{\sqrt{y}}\right) + x \cdot \log \left(\frac{\sqrt{x}}{\sqrt{y}}\right)\right) - z\\ \end{array}\]
x \cdot \log \left(\frac{x}{y}\right) - z
↓
\begin{array}{l}
\mathbf{if}\;y \le -5.330202515239102447628611255442331031352 \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 \log \left(\frac{\sqrt{x}}{\sqrt{y}}\right) + x \cdot \log \left(\frac{\sqrt{x}}{\sqrt{y}}\right)\right) - z\\

\end{array}
double f(double x, double y, double z) {
        double r89794537 = x;
        double r89794538 = y;
        double r89794539 = r89794537 / r89794538;
        double r89794540 = log(r89794539);
        double r89794541 = r89794537 * r89794540;
        double r89794542 = z;
        double r89794543 = r89794541 - r89794542;
        return r89794543;
}
↓
double f(double x, double y, double z) {
        double r89794544 = y;
        double r89794545 = -5.3302025152391e-309;
        bool r89794546 = r89794544 <= r89794545;
        double r89794547 = x;
        double r89794548 = -1.0;
        double r89794549 = r89794548 / r89794544;
        double r89794550 = log(r89794549);
        double r89794551 = r89794548 / r89794547;
        double r89794552 = log(r89794551);
        double r89794553 = r89794550 - r89794552;
        double r89794554 = r89794547 * r89794553;
        double r89794555 = z;
        double r89794556 = r89794554 - r89794555;
        double r89794557 = sqrt(r89794547);
        double r89794558 = sqrt(r89794544);
        double r89794559 = r89794557 / r89794558;
        double r89794560 = log(r89794559);
        double r89794561 = r89794547 * r89794560;
        double r89794562 = r89794561 + r89794561;
        double r89794563 = r89794562 - r89794555;
        double r89794564 = r89794546 ? r89794556 : r89794563;
        return r89794564;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. 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 -5.3302025152391e-309 < y

    1. Initial program 15.4

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

    3. Applied add-sqr-sqrt15.4

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

    4. Applied add-sqr-sqrt15.4

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

    5. Applied times-frac15.4

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

    6. Applied log-prod0.1

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

    7. Applied distribute-lft-in0.1

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2019173 
(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))