Average Error: 0.2 → 0.3
Time: 1.6s
Precision: binary64
\[\left(\left(x - \frac{16}{116}\right) \cdot 3\right) \cdot y\]
↓
\[\left(\sqrt{3} \cdot \left(\sqrt[3]{\sqrt{3}} \cdot \left(\left(x - 0.13793103448275862\right) \cdot \left(\sqrt[3]{\sqrt{3}} \cdot \sqrt[3]{\sqrt{3}}\right)\right)\right)\right) \cdot y\]
\left(\left(x - \frac{16}{116}\right) \cdot 3\right) \cdot y
↓
\left(\sqrt{3} \cdot \left(\sqrt[3]{\sqrt{3}} \cdot \left(\left(x - 0.13793103448275862\right) \cdot \left(\sqrt[3]{\sqrt{3}} \cdot \sqrt[3]{\sqrt{3}}\right)\right)\right)\right) \cdot y
(FPCore (x y) :precision binary64 (* (* (- x (/ 16.0 116.0)) 3.0) y))
↓
(FPCore (x y)
 :precision binary64
 (*
  (*
   (sqrt 3.0)
   (*
    (cbrt (sqrt 3.0))
    (* (- x 0.13793103448275862) (* (cbrt (sqrt 3.0)) (cbrt (sqrt 3.0))))))
  y))
double code(double x, double y) {
	return ((x - (16.0 / 116.0)) * 3.0) * y;
}
↓
double code(double x, double y) {
	return (sqrt(3.0) * (cbrt(sqrt(3.0)) * ((x - 0.13793103448275862) * (cbrt(sqrt(3.0)) * cbrt(sqrt(3.0)))))) * y;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original0.2
Target0.2
Herbie0.3
\[y \cdot \left(x \cdot 3 - 0.41379310344827586\right)\]

Derivation

  1. Initial program 0.2

    \[\left(\left(x - \frac{16}{116}\right) \cdot 3\right) \cdot y\]

  2. Simplified0.2

    \[\leadsto \color{blue}{\left(\left(x - 0.13793103448275862\right) \cdot 3\right) \cdot y}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt_binary641.0

    \[\leadsto \left(\left(x - 0.13793103448275862\right) \cdot \color{blue}{\left(\sqrt{3} \cdot \sqrt{3}\right)}\right) \cdot y\]

  5. Applied associate-*r*_binary640.9

    \[\leadsto \color{blue}{\left(\left(\left(x - 0.13793103448275862\right) \cdot \sqrt{3}\right) \cdot \sqrt{3}\right)} \cdot y\]
  6. Using strategy rm

  7. Applied add-cube-cbrt_binary640.2

    \[\leadsto \left(\left(\left(x - 0.13793103448275862\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt{3}} \cdot \sqrt[3]{\sqrt{3}}\right) \cdot \sqrt[3]{\sqrt{3}}\right)}\right) \cdot \sqrt{3}\right) \cdot y\]

  8. Applied associate-*r*_binary640.3

    \[\leadsto \left(\color{blue}{\left(\left(\left(x - 0.13793103448275862\right) \cdot \left(\sqrt[3]{\sqrt{3}} \cdot \sqrt[3]{\sqrt{3}}\right)\right) \cdot \sqrt[3]{\sqrt{3}}\right)} \cdot \sqrt{3}\right) \cdot y\]

  9. Final simplification0.3

    \[\leadsto \left(\sqrt{3} \cdot \left(\sqrt[3]{\sqrt{3}} \cdot \left(\left(x - 0.13793103448275862\right) \cdot \left(\sqrt[3]{\sqrt{3}} \cdot \sqrt[3]{\sqrt{3}}\right)\right)\right)\right) \cdot y\]

Reproduce

herbie shell --seed 2020220 
(FPCore (x y)
  :name "Data.Colour.CIE:cieLAB from colour-2.3.3, A"
  :precision binary64

  :herbie-target
  (* y (- (* x 3.0) 0.41379310344827586))

  (* (* (- x (/ 16.0 116.0)) 3.0) y))