Average Error: 0.4 → 0.4
Time: 4.8s
Precision: 64
\[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)\]
↓
\[\mathsf{fma}\left(\mathsf{fma}\left(2, \frac{1}{3}, -z \cdot 1\right) \cdot \left(y - x\right), 6, x\right) + \left(\left(y - x\right) \cdot 6\right) \cdot \mathsf{fma}\left(-z, 1, z \cdot 1\right)\]
x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)
↓
\mathsf{fma}\left(\mathsf{fma}\left(2, \frac{1}{3}, -z \cdot 1\right) \cdot \left(y - x\right), 6, x\right) + \left(\left(y - x\right) \cdot 6\right) \cdot \mathsf{fma}\left(-z, 1, z \cdot 1\right)
double f(double x, double y, double z) {
        double r191502 = x;
        double r191503 = y;
        double r191504 = r191503 - r191502;
        double r191505 = 6.0;
        double r191506 = r191504 * r191505;
        double r191507 = 2.0;
        double r191508 = 3.0;
        double r191509 = r191507 / r191508;
        double r191510 = z;
        double r191511 = r191509 - r191510;
        double r191512 = r191506 * r191511;
        double r191513 = r191502 + r191512;
        return r191513;
}
↓
double f(double x, double y, double z) {
        double r191514 = 2.0;
        double r191515 = 1.0;
        double r191516 = 3.0;
        double r191517 = r191515 / r191516;
        double r191518 = z;
        double r191519 = r191518 * r191515;
        double r191520 = -r191519;
        double r191521 = fma(r191514, r191517, r191520);
        double r191522 = y;
        double r191523 = x;
        double r191524 = r191522 - r191523;
        double r191525 = r191521 * r191524;
        double r191526 = 6.0;
        double r191527 = fma(r191525, r191526, r191523);
        double r191528 = r191524 * r191526;
        double r191529 = -r191518;
        double r191530 = fma(r191529, r191515, r191519);
        double r191531 = r191528 * r191530;
        double r191532 = r191527 + r191531;
        return r191532;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.4

    \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)\]
  2. Using strategy rm

  3. Applied *-un-lft-identity0.4

    \[\leadsto x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - \color{blue}{1 \cdot z}\right)\]

  4. Applied div-inv0.4

    \[\leadsto x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\color{blue}{2 \cdot \frac{1}{3}} - 1 \cdot z\right)\]

  5. Applied prod-diff0.4

    \[\leadsto x + \left(\left(y - x\right) \cdot 6\right) \cdot \color{blue}{\left(\mathsf{fma}\left(2, \frac{1}{3}, -z \cdot 1\right) + \mathsf{fma}\left(-z, 1, z \cdot 1\right)\right)}\]

  6. Applied distribute-lft-in0.4

    \[\leadsto x + \color{blue}{\left(\left(\left(y - x\right) \cdot 6\right) \cdot \mathsf{fma}\left(2, \frac{1}{3}, -z \cdot 1\right) + \left(\left(y - x\right) \cdot 6\right) \cdot \mathsf{fma}\left(-z, 1, z \cdot 1\right)\right)}\]

  7. Applied associate-+r+0.4

    \[\leadsto \color{blue}{\left(x + \left(\left(y - x\right) \cdot 6\right) \cdot \mathsf{fma}\left(2, \frac{1}{3}, -z \cdot 1\right)\right) + \left(\left(y - x\right) \cdot 6\right) \cdot \mathsf{fma}\left(-z, 1, z \cdot 1\right)}\]

  8. Simplified0.4

    \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(2, \frac{1}{3}, -z \cdot 1\right) \cdot \left(y - x\right), 6, x\right)} + \left(\left(y - x\right) \cdot 6\right) \cdot \mathsf{fma}\left(-z, 1, z \cdot 1\right)\]

  9. Final simplification0.4

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(2, \frac{1}{3}, -z \cdot 1\right) \cdot \left(y - x\right), 6, x\right) + \left(\left(y - x\right) \cdot 6\right) \cdot \mathsf{fma}\left(-z, 1, z \cdot 1\right)\]

Reproduce

herbie shell --seed 2020083 +o rules:numerics
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSL:hsl from colour-2.3.3, D"
  :precision binary64
  (+ x (* (* (- y x) 6) (- (/ 2 3) z))))