Average Error: 0.4 → 0.3
Time: 3.3s
Precision: 64
\[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)\]
↓
\[\mathsf{fma}\left(2 \cdot \frac{y - x}{3}, 6, x\right) + \left(\left(y - x\right) \cdot 6\right) \cdot \left(-z\right)\]
x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)
↓
\mathsf{fma}\left(2 \cdot \frac{y - x}{3}, 6, x\right) + \left(\left(y - x\right) \cdot 6\right) \cdot \left(-z\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((2.0 * ((y - x) / 3.0)), 6.0, x) + (((y - x) * 6.0) * -z));
}

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 sub-neg0.4

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

  4. Applied distribute-lft-in0.4

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

  5. Applied associate-+r+0.4

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

  6. Simplified0.4

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

  8. Applied div-inv0.4

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

  9. Applied associate-*l*0.4

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

  10. Simplified0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2020105 +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))))