Average Error: 0.1 → 0.1
Time: 21.7s
Precision: 64
\[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right)\]
↓
\[\mathsf{fma}\left(x, 0.5, y \cdot \left(\mathsf{fma}\left(2, \log \left(\sqrt[3]{z}\right), 1 - z\right) + \log \left(\sqrt[3]{z}\right)\right)\right)\]
x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right)
↓
\mathsf{fma}\left(x, 0.5, y \cdot \left(\mathsf{fma}\left(2, \log \left(\sqrt[3]{z}\right), 1 - z\right) + \log \left(\sqrt[3]{z}\right)\right)\right)
double f(double x, double y, double z) {
        double r232220 = x;
        double r232221 = 0.5;
        double r232222 = r232220 * r232221;
        double r232223 = y;
        double r232224 = 1.0;
        double r232225 = z;
        double r232226 = r232224 - r232225;
        double r232227 = log(r232225);
        double r232228 = r232226 + r232227;
        double r232229 = r232223 * r232228;
        double r232230 = r232222 + r232229;
        return r232230;
}
↓
double f(double x, double y, double z) {
        double r232231 = x;
        double r232232 = 0.5;
        double r232233 = y;
        double r232234 = 2.0;
        double r232235 = z;
        double r232236 = cbrt(r232235);
        double r232237 = log(r232236);
        double r232238 = 1.0;
        double r232239 = r232238 - r232235;
        double r232240 = fma(r232234, r232237, r232239);
        double r232241 = r232240 + r232237;
        double r232242 = r232233 * r232241;
        double r232243 = fma(r232231, r232232, r232242);
        return r232243;
}

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 add-cube-cbrt0.1

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

  5. Applied log-prod0.1

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

  6. Applied associate-+r+0.1

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

  7. Simplified0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2019303 +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)))))