Average Error: 0.1 → 0.1
Time: 28.8s
Precision: 64
\[x \cdot 0.5 + y \cdot \left(\left(1.0 - z\right) + \log z\right)\]
↓
\[\mathsf{fma}\left(x, 0.5, \left(1.0 - \left(\mathsf{fma}\left(-2, \log \left(\sqrt[3]{z}\right), z\right) - \log \left(\sqrt[3]{z}\right)\right)\right) \cdot y\right)\]
x \cdot 0.5 + y \cdot \left(\left(1.0 - z\right) + \log z\right)
↓
\mathsf{fma}\left(x, 0.5, \left(1.0 - \left(\mathsf{fma}\left(-2, \log \left(\sqrt[3]{z}\right), z\right) - \log \left(\sqrt[3]{z}\right)\right)\right) \cdot y\right)
double f(double x, double y, double z) {
        double r8608741 = x;
        double r8608742 = 0.5;
        double r8608743 = r8608741 * r8608742;
        double r8608744 = y;
        double r8608745 = 1.0;
        double r8608746 = z;
        double r8608747 = r8608745 - r8608746;
        double r8608748 = log(r8608746);
        double r8608749 = r8608747 + r8608748;
        double r8608750 = r8608744 * r8608749;
        double r8608751 = r8608743 + r8608750;
        return r8608751;
}
↓
double f(double x, double y, double z) {
        double r8608752 = x;
        double r8608753 = 0.5;
        double r8608754 = 1.0;
        double r8608755 = -2.0;
        double r8608756 = z;
        double r8608757 = cbrt(r8608756);
        double r8608758 = log(r8608757);
        double r8608759 = fma(r8608755, r8608758, r8608756);
        double r8608760 = r8608759 - r8608758;
        double r8608761 = r8608754 - r8608760;
        double r8608762 = y;
        double r8608763 = r8608761 * r8608762;
        double r8608764 = fma(r8608752, r8608753, r8608763);
        return r8608764;
}

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

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

  4. Applied add-cube-cbrt0.1

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

  5. Applied log-prod0.1

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

  6. Applied associate--r+0.1

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

  7. Simplified0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2019162 +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)))))