Average Error: 0.1 → 0.1
Time: 5.1s
Precision: 64
\[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right)\]
↓
\[x \cdot 0.5 + \left(\mathsf{fma}\left(y, 1 - z, y \cdot \log \left(\sqrt{z}\right)\right) + \log \left(\sqrt{z}\right) \cdot y\right)\]
x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right)
↓
x \cdot 0.5 + \left(\mathsf{fma}\left(y, 1 - z, y \cdot \log \left(\sqrt{z}\right)\right) + \log \left(\sqrt{z}\right) \cdot y\right)
double f(double x, double y, double z) {
        double r284291 = x;
        double r284292 = 0.5;
        double r284293 = r284291 * r284292;
        double r284294 = y;
        double r284295 = 1.0;
        double r284296 = z;
        double r284297 = r284295 - r284296;
        double r284298 = log(r284296);
        double r284299 = r284297 + r284298;
        double r284300 = r284294 * r284299;
        double r284301 = r284293 + r284300;
        return r284301;
}
↓
double f(double x, double y, double z) {
        double r284302 = x;
        double r284303 = 0.5;
        double r284304 = r284302 * r284303;
        double r284305 = y;
        double r284306 = 1.0;
        double r284307 = z;
        double r284308 = r284306 - r284307;
        double r284309 = sqrt(r284307);
        double r284310 = log(r284309);
        double r284311 = r284305 * r284310;
        double r284312 = fma(r284305, r284308, r284311);
        double r284313 = r284310 * r284305;
        double r284314 = r284312 + r284313;
        double r284315 = r284304 + r284314;
        return r284315;
}

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. Using strategy rm

  3. Applied distribute-lft-in0.1

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

  5. Applied add-sqr-sqrt0.1

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

  6. Applied log-prod0.1

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

  7. Applied distribute-rgt-in0.1

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

  8. Applied associate-+r+0.1

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

  9. Simplified0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2020081 +o rules:numerics
(FPCore (x y z)
  :name "System.Random.MWC.Distributions:gamma from mwc-random-0.13.3.2"
  :precision binary64

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

  (+ (* x 0.5) (* y (+ (- 1 z) (log z)))))