Average Error: 0.3 → 0.2
Time: 2.7s
Precision: binary64
\[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
↓
\[\mathsf{fma}\left(6 \cdot z, y - x, x\right) \]
x + \left(\left(y - x\right) \cdot 6\right) \cdot z
↓
\mathsf{fma}\left(6 \cdot z, y - x, x\right)

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

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original0.3
Target0.2
Herbie0.2
\[x - \left(6 \cdot z\right) \cdot \left(x - y\right) \]

Derivation

  1. Initial program 0.3

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

  2. Simplified0.3

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

  3. Applied fma-udef_binary640.3

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

  4. Simplified0.2

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

  5. Applied associate-*r*_binary640.2

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

  6. Applied fma-def_binary640.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2021275 
(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)))