Average Error: 0.1 → 0.1
Time: 24.9s
Precision: 64
\[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right)\]
↓
\[\mathsf{fma}\left(x, 0.5, \left(1 - \left(\left(\mathsf{fma}\left(-2, \log \left(\sqrt[3]{z}\right), z\right) - \log \left(\sqrt[3]{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right)\right) - \log \left(\sqrt[3]{\sqrt[3]{z}}\right)\right)\right) \cdot y\right)\]
x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right)
↓
\mathsf{fma}\left(x, 0.5, \left(1 - \left(\left(\mathsf{fma}\left(-2, \log \left(\sqrt[3]{z}\right), z\right) - \log \left(\sqrt[3]{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right)\right) - \log \left(\sqrt[3]{\sqrt[3]{z}}\right)\right)\right) \cdot y\right)
double f(double x, double y, double z) {
        double r12894577 = x;
        double r12894578 = 0.5;
        double r12894579 = r12894577 * r12894578;
        double r12894580 = y;
        double r12894581 = 1.0;
        double r12894582 = z;
        double r12894583 = r12894581 - r12894582;
        double r12894584 = log(r12894582);
        double r12894585 = r12894583 + r12894584;
        double r12894586 = r12894580 * r12894585;
        double r12894587 = r12894579 + r12894586;
        return r12894587;
}
↓
double f(double x, double y, double z) {
        double r12894588 = x;
        double r12894589 = 0.5;
        double r12894590 = 1.0;
        double r12894591 = -2.0;
        double r12894592 = z;
        double r12894593 = cbrt(r12894592);
        double r12894594 = log(r12894593);
        double r12894595 = fma(r12894591, r12894594, r12894592);
        double r12894596 = r12894593 * r12894593;
        double r12894597 = cbrt(r12894596);
        double r12894598 = log(r12894597);
        double r12894599 = r12894595 - r12894598;
        double r12894600 = cbrt(r12894593);
        double r12894601 = log(r12894600);
        double r12894602 = r12894599 - r12894601;
        double r12894603 = r12894590 - r12894602;
        double r12894604 = y;
        double r12894605 = r12894603 * r12894604;
        double r12894606 = fma(r12894588, r12894589, r12894605);
        return r12894606;
}

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

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

  4. Applied add-cube-cbrt0.1

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

  5. Applied log-prod0.1

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

  6. Applied associate--r+0.1

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

  7. Simplified0.1

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

  9. Applied add-cube-cbrt0.1

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

  10. Applied cbrt-prod0.1

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

  11. Applied log-prod0.1

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

  12. Applied associate--r+0.1

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

  13. Final simplification0.1

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

Reproduce

herbie shell --seed 2019169 +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)))))