Average Error: 0.3 → 0.2
Time: 2.3s
Precision: binary64
\[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
↓
\[x + \left(6 \cdot \left(y \cdot z\right) + -6 \cdot \left(z \cdot x\right)\right) \]
x + \left(\left(y - x\right) \cdot 6\right) \cdot z
↓
x + \left(6 \cdot \left(y \cdot z\right) + -6 \cdot \left(z \cdot x\right)\right)

(FPCore (x y z) :precision binary64 (+ x (* (* (- y x) 6.0) z)))
↓
(FPCore (x y z) :precision binary64 (+ x (+ (* 6.0 (* y z)) (* -6.0 (* z 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 x + ((6.0 * (y * z)) + (-6.0 * (z * 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

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. Taylor expanded in x around 0 0.2

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

  3. Simplified0.2

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

  4. Applied fma-udef_binary640.2

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

  5. Simplified0.3

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

  6. Applied sub-neg_binary640.3

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

  7. Applied distribute-rgt-in_binary640.3

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

  8. Applied distribute-rgt-in_binary640.3

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

  9. Simplified0.2

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

  10. Simplified0.2

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

  11. Final simplification0.2

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

Reproduce

herbie shell --seed 2021307 
(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)))