Average Error: 0.0 → 0.0
Time: 4.1s
Precision: 64
\[e^{\left(x + y \cdot \log y\right) - z}\]
↓
\[e^{\left(\mathsf{fma}\left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right), y, x\right) + y \cdot \log \left(\sqrt[3]{y}\right)\right) - z}\]
e^{\left(x + y \cdot \log y\right) - z}
↓
e^{\left(\mathsf{fma}\left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right), y, x\right) + y \cdot \log \left(\sqrt[3]{y}\right)\right) - z}
double f(double x, double y, double z) {
        double r309959 = x;
        double r309960 = y;
        double r309961 = log(r309960);
        double r309962 = r309960 * r309961;
        double r309963 = r309959 + r309962;
        double r309964 = z;
        double r309965 = r309963 - r309964;
        double r309966 = exp(r309965);
        return r309966;
}
↓
double f(double x, double y, double z) {
        double r309967 = y;
        double r309968 = cbrt(r309967);
        double r309969 = r309968 * r309968;
        double r309970 = log(r309969);
        double r309971 = x;
        double r309972 = fma(r309970, r309967, r309971);
        double r309973 = log(r309968);
        double r309974 = r309967 * r309973;
        double r309975 = r309972 + r309974;
        double r309976 = z;
        double r309977 = r309975 - r309976;
        double r309978 = exp(r309977);
        return r309978;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original0.0
Target0.0
Herbie0.0
\[e^{\left(x - z\right) + \log y \cdot y}\]

Derivation

  1. Initial program 0.0

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

  3. Applied add-cube-cbrt0.0

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

  4. Applied log-prod0.0

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

  5. Applied distribute-lft-in0.0

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

  6. Applied associate-+r+0.0

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

  7. Simplified0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2020062 +o rules:numerics
(FPCore (x y z)
  :name "Statistics.Distribution.Poisson.Internal:probability from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (exp (+ (- x z) (* (log y) y)))

  (exp (- (+ x (* y (log y))) z)))