Average Error: 0.4 → 0.1
Time: 2.9s
Precision: 64
\[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)\]
↓
\[\mathsf{fma}\left(y - x, \mathsf{fma}\left(1, 4, -z \cdot 6\right), \mathsf{fma}\left(\mathsf{fma}\left(-z, 6, z \cdot 6\right), y - x, x\right)\right)\]
x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)
↓
\mathsf{fma}\left(y - x, \mathsf{fma}\left(1, 4, -z \cdot 6\right), \mathsf{fma}\left(\mathsf{fma}\left(-z, 6, z \cdot 6\right), y - x, x\right)\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), fma(1.0, 4.0, -(z * 6.0)), fma(fma(-z, 6.0, (z * 6.0)), (y - x), x));
}

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. Simplified0.2

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

  3. Taylor expanded around 0 0.2

    \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{4 - 6 \cdot z}, x\right)\]
  4. Using strategy rm

  5. Applied fma-udef0.2

    \[\leadsto \color{blue}{\left(y - x\right) \cdot \left(4 - 6 \cdot z\right) + x}\]
  6. Using strategy rm

  7. Applied *-un-lft-identity0.2

    \[\leadsto \left(y - x\right) \cdot \left(\color{blue}{1 \cdot 4} - 6 \cdot z\right) + x\]

  8. Applied prod-diff0.2

    \[\leadsto \left(y - x\right) \cdot \color{blue}{\left(\mathsf{fma}\left(1, 4, -z \cdot 6\right) + \mathsf{fma}\left(-z, 6, z \cdot 6\right)\right)} + x\]

  9. Applied distribute-lft-in0.2

    \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot \mathsf{fma}\left(1, 4, -z \cdot 6\right) + \left(y - x\right) \cdot \mathsf{fma}\left(-z, 6, z \cdot 6\right)\right)} + x\]

  10. Applied associate-+l+0.2

    \[\leadsto \color{blue}{\left(y - x\right) \cdot \mathsf{fma}\left(1, 4, -z \cdot 6\right) + \left(\left(y - x\right) \cdot \mathsf{fma}\left(-z, 6, z \cdot 6\right) + x\right)}\]

  11. Simplified0.2

    \[\leadsto \left(y - x\right) \cdot \mathsf{fma}\left(1, 4, -z \cdot 6\right) + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-z, 6, z \cdot 6\right), y - x, x\right)}\]
  12. Using strategy rm

  13. Applied fma-def0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \mathsf{fma}\left(1, 4, -z \cdot 6\right), \mathsf{fma}\left(\mathsf{fma}\left(-z, 6, z \cdot 6\right), y - x, x\right)\right)}\]

  14. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(y - x, \mathsf{fma}\left(1, 4, -z \cdot 6\right), \mathsf{fma}\left(\mathsf{fma}\left(-z, 6, z \cdot 6\right), y - x, x\right)\right)\]

Reproduce

herbie shell --seed 2020056 +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))))