Average Error: 0.2 → 0.2
Time: 2.7s
Precision: binary64
\[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
↓
\[6 \cdot \left(y \cdot z\right) + \mathsf{fma}\left(x, z \cdot -6, x\right) \]
x + \left(\left(y - x\right) \cdot 6\right) \cdot z
↓
6 \cdot \left(y \cdot z\right) + \mathsf{fma}\left(x, z \cdot -6, x\right)

(FPCore (x y z) :precision binary64 (+ x (* (* (- y x) 6.0) z)))
↓
(FPCore (x y z) :precision binary64 (+ (* 6.0 (* y z)) (fma x (* z -6.0) x)))
double code(double x, double y, double z) {
	return x + (((y - x) * 6.0) * z);
}
↓
double code(double x, double y, double z) {
	return (6.0 * (y * z)) + fma(x, (z * -6.0), x);
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original0.2
Target0.2
Herbie0.2
\[x - \left(6 \cdot z\right) \cdot \left(x - y\right) \]

Derivation

  1. Initial program 0.2

    \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]

  2. Simplified0.2

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

  3. Taylor expanded in y around 0 0.2

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

  4. Applied associate--l+_binary640.2

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

  5. Simplified0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2022076 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSL:hsl from colour-2.3.3, E"
  :precision binary64

  :herbie-target
  (- x (* (* 6.0 z) (- x y)))

  (+ x (* (* (- y x) 6.0) z)))