Average Error: 0.4 → 0.2
Time: 4.0s
Precision: 64
\[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)\]
↓
\[\left(\left(y - x\right) \cdot \left(4 - 6 \cdot z\right) + \left(y - x\right) \cdot \mathsf{fma}\left(-z, 6, z \cdot 6\right)\right) + x\]
x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)
↓
\left(\left(y - x\right) \cdot \left(4 - 6 \cdot z\right) + \left(y - x\right) \cdot \mathsf{fma}\left(-z, 6, z \cdot 6\right)\right) + x
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 ((((y - x) * (4.0 - (6.0 * z))) + ((y - x) * fma(-z, 6.0, (z * 6.0)))) + 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 add-sqr-sqrt0.2

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

  8. Applied prod-diff0.2

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

  9. Applied distribute-lft-in0.2

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

  10. Simplified0.2

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

  11. Final simplification0.2

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

Reproduce

herbie shell --seed 2020100 +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))))