Average Error: 0.1 → 0.1
Time: 18.2s
Precision: 64
\[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right)\]
↓
\[\mathsf{fma}\left(y, \log \left(\sqrt{z}\right) + \left(1 - \left(z - \log \left(\sqrt{z}\right)\right)\right), 0.5 \cdot x\right)\]
x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right)
↓
\mathsf{fma}\left(y, \log \left(\sqrt{z}\right) + \left(1 - \left(z - \log \left(\sqrt{z}\right)\right)\right), 0.5 \cdot x\right)
double f(double x, double y, double z) {
        double r260610 = x;
        double r260611 = 0.5;
        double r260612 = r260610 * r260611;
        double r260613 = y;
        double r260614 = 1.0;
        double r260615 = z;
        double r260616 = r260614 - r260615;
        double r260617 = log(r260615);
        double r260618 = r260616 + r260617;
        double r260619 = r260613 * r260618;
        double r260620 = r260612 + r260619;
        return r260620;
}
↓
double f(double x, double y, double z) {
        double r260621 = y;
        double r260622 = z;
        double r260623 = sqrt(r260622);
        double r260624 = log(r260623);
        double r260625 = 1.0;
        double r260626 = r260622 - r260624;
        double r260627 = r260625 - r260626;
        double r260628 = r260624 + r260627;
        double r260629 = 0.5;
        double r260630 = x;
        double r260631 = r260629 * r260630;
        double r260632 = fma(r260621, r260628, r260631);
        return r260632;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original0.1
Target0.1
Herbie0.1
\[\left(y + 0.5 \cdot x\right) - y \cdot \left(z - \log z\right)\]

Derivation

  1. Initial program 0.1

    \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right)\]

  2. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(y, \log z + \left(1 - z\right), x \cdot 0.5\right)}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt0.1

    \[\leadsto \mathsf{fma}\left(y, \log \color{blue}{\left(\sqrt{z} \cdot \sqrt{z}\right)} + \left(1 - z\right), x \cdot 0.5\right)\]

  5. Applied log-prod0.1

    \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(\log \left(\sqrt{z}\right) + \log \left(\sqrt{z}\right)\right)} + \left(1 - z\right), x \cdot 0.5\right)\]

  6. Applied associate-+l+0.1

    \[\leadsto \mathsf{fma}\left(y, \color{blue}{\log \left(\sqrt{z}\right) + \left(\log \left(\sqrt{z}\right) + \left(1 - z\right)\right)}, x \cdot 0.5\right)\]

  7. Simplified0.1

    \[\leadsto \mathsf{fma}\left(y, \log \left(\sqrt{z}\right) + \color{blue}{\left(1 - \left(z - \log \left(\sqrt{z}\right)\right)\right)}, x \cdot 0.5\right)\]

  8. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(y, \log \left(\sqrt{z}\right) + \left(1 - \left(z - \log \left(\sqrt{z}\right)\right)\right), 0.5 \cdot x\right)\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x y z)
  :name "System.Random.MWC.Distributions:gamma from mwc-random-0.13.3.2"

  :herbie-target
  (- (+ y (* 0.5 x)) (* y (- z (log z))))

  (+ (* x 0.5) (* y (+ (- 1.0 z) (log z)))))