Average Error: 44.9 → 44.9
Time: 4.8s
Precision: 64
\[\mathsf{fma}\left(x, y, z\right) - \left(1 + \left(x \cdot y + z\right)\right)\]
↓
\[\left(\sqrt[3]{\mathsf{fma}\left(x, y, z\right) - \left(1 + \left(x \cdot y + z\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x, y, z\right) - \left(1 + \left(x \cdot y + z\right)\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(x, y, z\right) - \left(1 + \left(x \cdot y + z\right)\right)}\]
\mathsf{fma}\left(x, y, z\right) - \left(1 + \left(x \cdot y + z\right)\right)
↓
\left(\sqrt[3]{\mathsf{fma}\left(x, y, z\right) - \left(1 + \left(x \cdot y + z\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x, y, z\right) - \left(1 + \left(x \cdot y + z\right)\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(x, y, z\right) - \left(1 + \left(x \cdot y + z\right)\right)}
double f(double x, double y, double z) {
        double r64471 = x;
        double r64472 = y;
        double r64473 = z;
        double r64474 = fma(r64471, r64472, r64473);
        double r64475 = 1.0;
        double r64476 = r64471 * r64472;
        double r64477 = r64476 + r64473;
        double r64478 = r64475 + r64477;
        double r64479 = r64474 - r64478;
        return r64479;
}
↓
double f(double x, double y, double z) {
        double r64480 = x;
        double r64481 = y;
        double r64482 = z;
        double r64483 = fma(r64480, r64481, r64482);
        double r64484 = 1.0;
        double r64485 = r64480 * r64481;
        double r64486 = r64485 + r64482;
        double r64487 = r64484 + r64486;
        double r64488 = r64483 - r64487;
        double r64489 = cbrt(r64488);
        double r64490 = r64489 * r64489;
        double r64491 = r64490 * r64489;
        return r64491;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original44.9
Target0
Herbie44.9
\[-1\]

Derivation

  1. Initial program 44.9

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

  3. Applied add-cube-cbrt44.9

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

  4. Final simplification44.9

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

Reproduce

herbie shell --seed 2020047 
(FPCore (x y z)
  :name "simple fma test"
  :precision binary64

  :herbie-target
  -1

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