Average Error: 15.2 → 0.2
Time: 5.7s
Precision: 64
\[x \cdot \log \left(\frac{x}{y}\right) - z\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.0411648480087909 \cdot 10^{-94}:\\ \;\;\;\;\left(\left(\left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) \cdot 1\right) \cdot x + x \cdot \log \left(\frac{{\left(-1 \cdot x\right)}^{\frac{1}{3}} \cdot \sqrt[3]{-1}}{\sqrt[3]{y}}\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) \cdot 1\right) \cdot x + x \cdot \log \left(\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{x}\right)\right)}{\sqrt[3]{y}}\right)\right) - z\\ \end{array}\]
x \cdot \log \left(\frac{x}{y}\right) - z
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.0411648480087909 \cdot 10^{-94}:\\
\;\;\;\;\left(\left(\left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) \cdot 1\right) \cdot x + x \cdot \log \left(\frac{{\left(-1 \cdot x\right)}^{\frac{1}{3}} \cdot \sqrt[3]{-1}}{\sqrt[3]{y}}\right)\right) - z\\

\mathbf{else}:\\
\;\;\;\;\left(\left(\left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) \cdot 1\right) \cdot x + x \cdot \log \left(\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{x}\right)\right)}{\sqrt[3]{y}}\right)\right) - z\\

\end{array}
double f(double x, double y, double z) {
        double r535516 = x;
        double r535517 = y;
        double r535518 = r535516 / r535517;
        double r535519 = log(r535518);
        double r535520 = r535516 * r535519;
        double r535521 = z;
        double r535522 = r535520 - r535521;
        return r535522;
}
↓
double f(double x, double y, double z) {
        double r535523 = x;
        double r535524 = -1.0411648480087909e-94;
        bool r535525 = r535523 <= r535524;
        double r535526 = 2.0;
        double r535527 = cbrt(r535523);
        double r535528 = y;
        double r535529 = cbrt(r535528);
        double r535530 = r535527 / r535529;
        double r535531 = log(r535530);
        double r535532 = r535526 * r535531;
        double r535533 = 1.0;
        double r535534 = r535532 * r535533;
        double r535535 = r535534 * r535523;
        double r535536 = -1.0;
        double r535537 = r535536 * r535523;
        double r535538 = 0.3333333333333333;
        double r535539 = pow(r535537, r535538);
        double r535540 = cbrt(r535536);
        double r535541 = r535539 * r535540;
        double r535542 = r535541 / r535529;
        double r535543 = log(r535542);
        double r535544 = r535523 * r535543;
        double r535545 = r535535 + r535544;
        double r535546 = z;
        double r535547 = r535545 - r535546;
        double r535548 = log1p(r535527);
        double r535549 = expm1(r535548);
        double r535550 = r535549 / r535529;
        double r535551 = log(r535550);
        double r535552 = r535523 * r535551;
        double r535553 = r535535 + r535552;
        double r535554 = r535553 - r535546;
        double r535555 = r535525 ? r535547 : r535554;
        return r535555;
}

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

Original15.2
Target7.9
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;y \lt 7.59507779908377277 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -1.0411648480087909e-94

    1. Initial program 13.1

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

    3. Applied add-cube-cbrt13.1

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

    4. Applied add-cube-cbrt13.1

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

    5. Applied times-frac13.1

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

    6. Applied log-prod2.9

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

    7. Applied distribute-lft-in2.9

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

    8. Simplified0.3

      \[\leadsto \left(\color{blue}{\left(\left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) \cdot 1\right) \cdot x} + x \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) - z\]

    9. Taylor expanded around -inf 0.3

      \[\leadsto \left(\left(\left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) \cdot 1\right) \cdot x + x \cdot \log \left(\frac{\color{blue}{{\left(-1 \cdot x\right)}^{\frac{1}{3}} \cdot \sqrt[3]{-1}}}{\sqrt[3]{y}}\right)\right) - z\]

    if -1.0411648480087909e-94 < x

    1. Initial program 16.2

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

    3. Applied add-cube-cbrt16.2

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

    4. Applied add-cube-cbrt16.2

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

    5. Applied times-frac16.2

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

    6. Applied log-prod3.9

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

    7. Applied distribute-lft-in3.9

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

    8. Simplified0.2

      \[\leadsto \left(\color{blue}{\left(\left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) \cdot 1\right) \cdot x} + x \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) - z\]
    9. Using strategy rm

    10. Applied expm1-log1p-u0.2

      \[\leadsto \left(\left(\left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) \cdot 1\right) \cdot x + x \cdot \log \left(\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{x}\right)\right)}}{\sqrt[3]{y}}\right)\right) - z\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0411648480087909 \cdot 10^{-94}:\\ \;\;\;\;\left(\left(\left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) \cdot 1\right) \cdot x + x \cdot \log \left(\frac{{\left(-1 \cdot x\right)}^{\frac{1}{3}} \cdot \sqrt[3]{-1}}{\sqrt[3]{y}}\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) \cdot 1\right) \cdot x + x \cdot \log \left(\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{x}\right)\right)}{\sqrt[3]{y}}\right)\right) - z\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 +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))