Average Error: 0.4 → 0.2
Time: 10.0s
Precision: 64
\[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)\]
↓
\[\mathsf{fma}\left(y - x, 6 \cdot \left(\frac{2}{3} - z\right), x\right) + \mathsf{fma}\left(-z, 1, z \cdot 1\right) \cdot \left(\left(y - x\right) \cdot 6\right)\]
x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)
↓
\mathsf{fma}\left(y - x, 6 \cdot \left(\frac{2}{3} - z\right), x\right) + \mathsf{fma}\left(-z, 1, z \cdot 1\right) \cdot \left(\left(y - x\right) \cdot 6\right)
double f(double x, double y, double z) {
        double r1321 = x;
        double r1322 = y;
        double r1323 = r1322 - r1321;
        double r1324 = 6.0;
        double r1325 = r1323 * r1324;
        double r1326 = 2.0;
        double r1327 = 3.0;
        double r1328 = r1326 / r1327;
        double r1329 = z;
        double r1330 = r1328 - r1329;
        double r1331 = r1325 * r1330;
        double r1332 = r1321 + r1331;
        return r1332;
}
↓
double f(double x, double y, double z) {
        double r1333 = y;
        double r1334 = x;
        double r1335 = r1333 - r1334;
        double r1336 = 6.0;
        double r1337 = 2.0;
        double r1338 = 3.0;
        double r1339 = r1337 / r1338;
        double r1340 = z;
        double r1341 = r1339 - r1340;
        double r1342 = r1336 * r1341;
        double r1343 = fma(r1335, r1342, r1334);
        double r1344 = -r1340;
        double r1345 = 1.0;
        double r1346 = r1340 * r1345;
        double r1347 = fma(r1344, r1345, r1346);
        double r1348 = r1335 * r1336;
        double r1349 = r1347 * r1348;
        double r1350 = r1343 + r1349;
        return r1350;
}

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 add-sqr-sqrt0.4

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

  5. Applied prod-diff0.4

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

  6. Applied distribute-rgt-in0.4

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

  7. Applied associate-+r+0.4

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

  8. Simplified0.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2020025 +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))))