Average Error: 0.1 → 0.1
Time: 4.3s
Precision: 64
\[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right)\]
↓
\[x \cdot 0.5 + \mathsf{fma}\left(1 - \left(z - 2 \cdot \log \left(\sqrt[3]{z}\right)\right), y, \log \left(\sqrt[3]{z}\right) \cdot y\right)\]
x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right)
↓
x \cdot 0.5 + \mathsf{fma}\left(1 - \left(z - 2 \cdot \log \left(\sqrt[3]{z}\right)\right), y, \log \left(\sqrt[3]{z}\right) \cdot y\right)
double f(double x, double y, double z) {
        double r253250 = x;
        double r253251 = 0.5;
        double r253252 = r253250 * r253251;
        double r253253 = y;
        double r253254 = 1.0;
        double r253255 = z;
        double r253256 = r253254 - r253255;
        double r253257 = log(r253255);
        double r253258 = r253256 + r253257;
        double r253259 = r253253 * r253258;
        double r253260 = r253252 + r253259;
        return r253260;
}
↓
double f(double x, double y, double z) {
        double r253261 = x;
        double r253262 = 0.5;
        double r253263 = r253261 * r253262;
        double r253264 = 1.0;
        double r253265 = z;
        double r253266 = 2.0;
        double r253267 = cbrt(r253265);
        double r253268 = log(r253267);
        double r253269 = r253266 * r253268;
        double r253270 = r253265 - r253269;
        double r253271 = r253264 - r253270;
        double r253272 = y;
        double r253273 = r253268 * r253272;
        double r253274 = fma(r253271, r253272, r253273);
        double r253275 = r253263 + r253274;
        return r253275;
}

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

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

  4. Applied fma-udef0.1

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

  6. Applied distribute-lft-in0.1

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

  8. Applied add-cube-cbrt0.1

    \[\leadsto x \cdot 0.5 + \left(y \cdot \left(1 - z\right) + y \cdot \log \color{blue}{\left(\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}\right)}\right)\]

  9. Applied log-prod0.1

    \[\leadsto x \cdot 0.5 + \left(y \cdot \left(1 - z\right) + y \cdot \color{blue}{\left(\log \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) + \log \left(\sqrt[3]{z}\right)\right)}\right)\]

  10. Applied distribute-rgt-in0.1

    \[\leadsto x \cdot 0.5 + \left(y \cdot \left(1 - z\right) + \color{blue}{\left(\log \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot y + \log \left(\sqrt[3]{z}\right) \cdot y\right)}\right)\]

  11. Applied associate-+r+0.1

    \[\leadsto x \cdot 0.5 + \color{blue}{\left(\left(y \cdot \left(1 - z\right) + \log \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot y\right) + \log \left(\sqrt[3]{z}\right) \cdot y\right)}\]

  12. Simplified0.1

    \[\leadsto x \cdot 0.5 + \left(\color{blue}{\left(1 - \left(z - 2 \cdot \log \left(\sqrt[3]{z}\right)\right)\right) \cdot y} + \log \left(\sqrt[3]{z}\right) \cdot y\right)\]
  13. Using strategy rm

  14. Applied fma-def0.1

    \[\leadsto x \cdot 0.5 + \color{blue}{\mathsf{fma}\left(1 - \left(z - 2 \cdot \log \left(\sqrt[3]{z}\right)\right), y, \log \left(\sqrt[3]{z}\right) \cdot y\right)}\]

  15. Final simplification0.1

    \[\leadsto x \cdot 0.5 + \mathsf{fma}\left(1 - \left(z - 2 \cdot \log \left(\sqrt[3]{z}\right)\right), y, \log \left(\sqrt[3]{z}\right) \cdot y\right)\]

Reproduce

herbie shell --seed 2020039 +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)))))