Average Error: 0.1 → 0.1
Time: 23.2s
Precision: 64
\[\left(x \cdot \log y - z\right) - y\]
↓
\[\left(\mathsf{fma}\left(x, 2 \cdot \log \left(\sqrt[3]{y}\right), \log \left(\sqrt[3]{{y}^{\frac{2}{3}}}\right) \cdot x + x \cdot \log \left(\sqrt[3]{\sqrt[3]{y}}\right)\right) - z\right) - y\]
\left(x \cdot \log y - z\right) - y
↓
\left(\mathsf{fma}\left(x, 2 \cdot \log \left(\sqrt[3]{y}\right), \log \left(\sqrt[3]{{y}^{\frac{2}{3}}}\right) \cdot x + x \cdot \log \left(\sqrt[3]{\sqrt[3]{y}}\right)\right) - z\right) - y
double f(double x, double y, double z) {
        double r31248 = x;
        double r31249 = y;
        double r31250 = log(r31249);
        double r31251 = r31248 * r31250;
        double r31252 = z;
        double r31253 = r31251 - r31252;
        double r31254 = r31253 - r31249;
        return r31254;
}
↓
double f(double x, double y, double z) {
        double r31255 = x;
        double r31256 = 2.0;
        double r31257 = y;
        double r31258 = cbrt(r31257);
        double r31259 = log(r31258);
        double r31260 = r31256 * r31259;
        double r31261 = 0.6666666666666666;
        double r31262 = pow(r31257, r31261);
        double r31263 = cbrt(r31262);
        double r31264 = log(r31263);
        double r31265 = r31264 * r31255;
        double r31266 = cbrt(r31258);
        double r31267 = log(r31266);
        double r31268 = r31255 * r31267;
        double r31269 = r31265 + r31268;
        double r31270 = fma(r31255, r31260, r31269);
        double r31271 = z;
        double r31272 = r31270 - r31271;
        double r31273 = r31272 - r31257;
        return r31273;
}

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-cbrt0.1

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

  4. Applied log-prod0.1

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

  5. Applied distribute-lft-in0.1

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

  6. Simplified0.1

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

  8. Applied fma-def0.1

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

  10. Applied add-cube-cbrt0.1

    \[\leadsto \left(\mathsf{fma}\left(x, 2 \cdot \log \left(\sqrt[3]{y}\right), x \cdot \log \left(\sqrt[3]{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}\right)\right) - z\right) - y\]

  11. Applied cbrt-prod0.1

    \[\leadsto \left(\mathsf{fma}\left(x, 2 \cdot \log \left(\sqrt[3]{y}\right), x \cdot \log \color{blue}{\left(\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right)}\right) - z\right) - y\]

  12. Applied log-prod0.1

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

  13. Applied distribute-lft-in0.1

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

  14. Simplified0.1

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

  15. Final simplification0.1

    \[\leadsto \left(\mathsf{fma}\left(x, 2 \cdot \log \left(\sqrt[3]{y}\right), \log \left(\sqrt[3]{{y}^{\frac{2}{3}}}\right) \cdot x + x \cdot \log \left(\sqrt[3]{\sqrt[3]{y}}\right)\right) - z\right) - y\]

Reproduce

herbie shell --seed 2019208 +o rules:numerics
(FPCore (x y z)
  :name "Statistics.Distribution.Poisson:$clogProbability from math-functions-0.1.5.2"
  :precision binary64
  (- (- (* x (log y)) z) y))