Average Error: 0.1 → 0.1
Time: 30.5s
Precision: 64
\[x \cdot 0.5 + y \cdot \left(\left(1.0 - z\right) + \log z\right)\]
↓
\[\mathsf{fma}\left(y, \log \left({z}^{\frac{2}{3}}\right) + \left(\log \left(\sqrt[3]{z}\right) - \left(z - 1.0\right)\right), 0.5 \cdot x\right)\]
x \cdot 0.5 + y \cdot \left(\left(1.0 - z\right) + \log z\right)
↓
\mathsf{fma}\left(y, \log \left({z}^{\frac{2}{3}}\right) + \left(\log \left(\sqrt[3]{z}\right) - \left(z - 1.0\right)\right), 0.5 \cdot x\right)
double f(double x, double y, double z) {
        double r12710206 = x;
        double r12710207 = 0.5;
        double r12710208 = r12710206 * r12710207;
        double r12710209 = y;
        double r12710210 = 1.0;
        double r12710211 = z;
        double r12710212 = r12710210 - r12710211;
        double r12710213 = log(r12710211);
        double r12710214 = r12710212 + r12710213;
        double r12710215 = r12710209 * r12710214;
        double r12710216 = r12710208 + r12710215;
        return r12710216;
}
↓
double f(double x, double y, double z) {
        double r12710217 = y;
        double r12710218 = z;
        double r12710219 = 0.6666666666666666;
        double r12710220 = pow(r12710218, r12710219);
        double r12710221 = log(r12710220);
        double r12710222 = cbrt(r12710218);
        double r12710223 = log(r12710222);
        double r12710224 = 1.0;
        double r12710225 = r12710218 - r12710224;
        double r12710226 = r12710223 - r12710225;
        double r12710227 = r12710221 + r12710226;
        double r12710228 = 0.5;
        double r12710229 = x;
        double r12710230 = r12710228 * r12710229;
        double r12710231 = fma(r12710217, r12710227, r12710230);
        return r12710231;
}

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

    \[x \cdot 0.5 + y \cdot \left(\left(1.0 - z\right) + \log z\right)\]

  2. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(y, \log z - \left(z - 1.0\right), 0.5 \cdot x\right)}\]
  3. Using strategy rm

  4. Applied add-cube-cbrt0.1

    \[\leadsto \mathsf{fma}\left(y, \log \color{blue}{\left(\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}\right)} - \left(z - 1.0\right), 0.5 \cdot x\right)\]

  5. Applied log-prod0.1

    \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(\log \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) + \log \left(\sqrt[3]{z}\right)\right)} - \left(z - 1.0\right), 0.5 \cdot x\right)\]

  6. Applied associate--l+0.1

    \[\leadsto \mathsf{fma}\left(y, \color{blue}{\log \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) + \left(\log \left(\sqrt[3]{z}\right) - \left(z - 1.0\right)\right)}, 0.5 \cdot x\right)\]
  7. Using strategy rm

  8. Applied pow1/30.1

    \[\leadsto \mathsf{fma}\left(y, \log \left(\sqrt[3]{z} \cdot \color{blue}{{z}^{\frac{1}{3}}}\right) + \left(\log \left(\sqrt[3]{z}\right) - \left(z - 1.0\right)\right), 0.5 \cdot x\right)\]

  9. Applied pow1/30.1

    \[\leadsto \mathsf{fma}\left(y, \log \left(\color{blue}{{z}^{\frac{1}{3}}} \cdot {z}^{\frac{1}{3}}\right) + \left(\log \left(\sqrt[3]{z}\right) - \left(z - 1.0\right)\right), 0.5 \cdot x\right)\]

  10. Applied pow-prod-up0.1

    \[\leadsto \mathsf{fma}\left(y, \log \color{blue}{\left({z}^{\left(\frac{1}{3} + \frac{1}{3}\right)}\right)} + \left(\log \left(\sqrt[3]{z}\right) - \left(z - 1.0\right)\right), 0.5 \cdot x\right)\]

  11. Simplified0.1

    \[\leadsto \mathsf{fma}\left(y, \log \left({z}^{\color{blue}{\frac{2}{3}}}\right) + \left(\log \left(\sqrt[3]{z}\right) - \left(z - 1.0\right)\right), 0.5 \cdot x\right)\]

  12. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(y, \log \left({z}^{\frac{2}{3}}\right) + \left(\log \left(\sqrt[3]{z}\right) - \left(z - 1.0\right)\right), 0.5 \cdot x\right)\]

Reproduce

herbie shell --seed 2019163 +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)))))