Average Error: 0.4 → 0.3
Time: 4.7s
Precision: 64
\[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)\]
↓
\[\mathsf{fma}\left(y - x, 6 \cdot \left(\frac{2}{3} - z\right), x\right) + \left(\left(y - x\right) \cdot 6\right) \cdot \mathsf{fma}\left(-z, 1, z \cdot 1\right)\]
x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)
↓
\mathsf{fma}\left(y - x, 6 \cdot \left(\frac{2}{3} - z\right), x\right) + \left(\left(y - x\right) \cdot 6\right) \cdot \mathsf{fma}\left(-z, 1, z \cdot 1\right)
double code(double x, double y, double z) {
	return (x + (((y - x) * 6.0) * ((2.0 / 3.0) - z)));
}
↓
double code(double x, double y, double z) {
	return (fma((y - x), (6.0 * ((2.0 / 3.0) - z)), x) + (((y - x) * 6.0) * fma(-z, 1.0, (z * 1.0))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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-lft-in0.4

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

  7. Applied associate-+r+0.4

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

  8. Simplified0.3

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

  9. Final simplification0.3

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

Reproduce

herbie shell --seed 2020078 +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))))