Average Error: 0.1 → 0.1
Time: 15.6s
Precision: 64
\[\left(x \cdot \log y - z\right) - y\]
↓
\[\mathsf{fma}\left(y, -1, y\right) + \mathsf{fma}\left(x, \log y, -\left(y + z\right)\right)\]
\left(x \cdot \log y - z\right) - y
↓
\mathsf{fma}\left(y, -1, y\right) + \mathsf{fma}\left(x, \log y, -\left(y + z\right)\right)
double f(double x, double y, double z) {
        double r27862 = x;
        double r27863 = y;
        double r27864 = log(r27863);
        double r27865 = r27862 * r27864;
        double r27866 = z;
        double r27867 = r27865 - r27866;
        double r27868 = r27867 - r27863;
        return r27868;
}
↓
double f(double x, double y, double z) {
        double r27869 = y;
        double r27870 = -1.0;
        double r27871 = fma(r27869, r27870, r27869);
        double r27872 = x;
        double r27873 = log(r27869);
        double r27874 = z;
        double r27875 = r27869 + r27874;
        double r27876 = -r27875;
        double r27877 = fma(r27872, r27873, r27876);
        double r27878 = r27871 + r27877;
        return r27878;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.1

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

  3. Applied add-cube-cbrt0.5

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

  4. Applied add-sqr-sqrt32.8

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

  5. Applied prod-diff32.8

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

  6. Simplified0.1

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

  7. Simplified0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x y z)
  :name "Statistics.Distribution.Poisson:$clogProbability from math-functions-0.1.5.2"
  (- (- (* x (log y)) z) y))