Average Error: 45.2 → 8.0
Time: 3.5s
Precision: 64
\[\mathsf{fma}\left(x, y, z\right) - \left(1 + \left(x \cdot y + z\right)\right)\]
↓
\[\sqrt[3]{{\left(\left(\mathsf{fma}\left(x, y, z\right) + \left(-\left(z + x \cdot y\right)\right)\right) - 1\right)}^{3}}\]
\mathsf{fma}\left(x, y, z\right) - \left(1 + \left(x \cdot y + z\right)\right)
↓
\sqrt[3]{{\left(\left(\mathsf{fma}\left(x, y, z\right) + \left(-\left(z + x \cdot y\right)\right)\right) - 1\right)}^{3}}
double f(double x, double y, double z) {
        double r76462 = x;
        double r76463 = y;
        double r76464 = z;
        double r76465 = fma(r76462, r76463, r76464);
        double r76466 = 1.0;
        double r76467 = r76462 * r76463;
        double r76468 = r76467 + r76464;
        double r76469 = r76466 + r76468;
        double r76470 = r76465 - r76469;
        return r76470;
}
↓
double f(double x, double y, double z) {
        double r76471 = x;
        double r76472 = y;
        double r76473 = z;
        double r76474 = fma(r76471, r76472, r76473);
        double r76475 = r76471 * r76472;
        double r76476 = r76473 + r76475;
        double r76477 = -r76476;
        double r76478 = r76474 + r76477;
        double r76479 = 1.0;
        double r76480 = r76478 - r76479;
        double r76481 = 3.0;
        double r76482 = pow(r76480, r76481);
        double r76483 = cbrt(r76482);
        return r76483;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original45.2
Target0
Herbie8.0
\[-1\]

Derivation

  1. Initial program 45.2

    \[\mathsf{fma}\left(x, y, z\right) - \left(1 + \left(x \cdot y + z\right)\right)\]
  2. Using strategy rm

  3. Applied add-cbrt-cube45.2

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

  4. Simplified45.2

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

  6. Applied associate--r+30.9

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

  8. Applied associate--r+14.7

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

  10. Applied sub-neg14.7

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

  11. Applied associate--l+8.0

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

  12. Simplified8.0

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

  13. Final simplification8.0

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

Reproduce

herbie shell --seed 2019354 
(FPCore (x y z)
  :name "simple fma test"
  :precision binary64

  :herbie-target
  -1

  (- (fma x y z) (+ 1 (+ (* x y) z))))