Average Error: 0.0 → 0.0
Time: 3.7s
Precision: 64
\[e^{\left(x + y \cdot \log y\right) - z}\]
↓
\[e^{\left(\mathsf{fma}\left(\log \left(\sqrt{y}\right), y, x\right) + \log \left(\sqrt{y}\right) \cdot y\right) - z}\]
e^{\left(x + y \cdot \log y\right) - z}
↓
e^{\left(\mathsf{fma}\left(\log \left(\sqrt{y}\right), y, x\right) + \log \left(\sqrt{y}\right) \cdot y\right) - z}
double f(double x, double y, double z) {
        double r260570 = x;
        double r260571 = y;
        double r260572 = log(r260571);
        double r260573 = r260571 * r260572;
        double r260574 = r260570 + r260573;
        double r260575 = z;
        double r260576 = r260574 - r260575;
        double r260577 = exp(r260576);
        return r260577;
}
↓
double f(double x, double y, double z) {
        double r260578 = y;
        double r260579 = sqrt(r260578);
        double r260580 = log(r260579);
        double r260581 = x;
        double r260582 = fma(r260580, r260578, r260581);
        double r260583 = r260580 * r260578;
        double r260584 = r260582 + r260583;
        double r260585 = z;
        double r260586 = r260584 - r260585;
        double r260587 = exp(r260586);
        return r260587;
}

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-sqr-sqrt0.0

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

  4. Applied log-prod0.0

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

  5. Applied distribute-rgt-in0.0

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

  6. Applied associate-+r+0.0

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

  7. Simplified0.0

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

  8. Final simplification0.0

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

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)))