Average Error: 0.3 → 0.2
Time: 3.5s
Precision: binary64
\[x + \left(\left(y - x\right) \cdot 6\right) \cdot z\]
↓
\[x \cdot \left(1 - 6 \cdot z\right) + 6 \cdot \left(z \cdot y\right)\]
x + \left(\left(y - x\right) \cdot 6\right) \cdot z
↓
x \cdot \left(1 - 6 \cdot z\right) + 6 \cdot \left(z \cdot y\right)
(FPCore (x y z) :precision binary64 (+ x (* (* (- y x) 6.0) z)))
↓
(FPCore (x y z)
 :precision binary64
 (+ (* x (- 1.0 (* 6.0 z))) (* 6.0 (* z y))))
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 * (1.0 - (6.0 * z))) + (6.0 * (z * y));
}

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 around 0 0.2

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

  3. Simplified0.2

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

  5. Applied *-un-lft-identity_binary640.2

    \[\leadsto x + -6 \cdot \left(z \cdot \left(x - \color{blue}{1 \cdot y}\right)\right)\]

  6. Applied cancel-sign-sub-inv_binary640.2

    \[\leadsto x + -6 \cdot \left(z \cdot \color{blue}{\left(x + \left(-1\right) \cdot y\right)}\right)\]

  7. Applied distribute-rgt-in_binary640.2

    \[\leadsto x + -6 \cdot \color{blue}{\left(x \cdot z + \left(\left(-1\right) \cdot y\right) \cdot z\right)}\]

  8. Applied distribute-lft-in_binary640.2

    \[\leadsto x + \color{blue}{\left(-6 \cdot \left(x \cdot z\right) + -6 \cdot \left(\left(\left(-1\right) \cdot y\right) \cdot z\right)\right)}\]

  9. Applied associate-+r+_binary640.2

    \[\leadsto \color{blue}{\left(x + -6 \cdot \left(x \cdot z\right)\right) + -6 \cdot \left(\left(\left(-1\right) \cdot y\right) \cdot z\right)}\]

  10. Simplified0.2

    \[\leadsto \color{blue}{x \cdot \left(1 - 6 \cdot z\right)} + -6 \cdot \left(\left(\left(-1\right) \cdot y\right) \cdot z\right)\]

  11. Final simplification0.2

    \[\leadsto x \cdot \left(1 - 6 \cdot z\right) + 6 \cdot \left(z \cdot y\right)\]

Alternatives

Reproduce

herbie shell --seed 2021118 
(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)))