Average Error: 0.2 → 0.2
Time: 2.1s
Precision: binary64
\[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
↓
\[\mathsf{fma}\left(6, z \cdot y - z \cdot x, x\right) \]
x + \left(\left(y - x\right) \cdot 6\right) \cdot z
↓
\mathsf{fma}\left(6, z \cdot y - z \cdot 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) (* z 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) - (z * x)), 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. Simplified0.2

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

  5. Applied sub-neg_binary640.2

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

  6. Applied distribute-lft-in_binary640.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2021313 
(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)))